## Appendix B: Geometry

Using properties of angles to solve problems, learning outcomes.

- Find the supplement of an angle
- Find the complement of an angle

Are you familiar with the phrase ‘do a [latex]180[/latex]?’ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[/latex] degrees. See the image below.

[latex]\angle A[/latex] is the angle with vertex at [latex]\text{point }A[/latex].

We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] to for the measure of an angle. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ [/latex], we would write [latex]m\angle A=27[/latex].

If the sum of the measures of two angles is [latex]\text{180}^ \circ[/latex], then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to [latex]\text{180}^ \circ [/latex]. Each angle is the supplement of the other.

The sum of the measures of supplementary angles is [latex]\text{180}^ \circ [/latex].

The sum of the measures of complementary angles is [latex]\text{90}^ \circ[/latex].

## Supplementary and Complementary Angles

If the sum of the measures of two angles is [latex]\text{180}^\circ [/latex], then the angles are supplementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are supplementary, then [latex]m\angle{A}+m\angle{B}=180^\circ[/latex].

If the sum of the measures of two angles is [latex]\text{90}^\circ[/latex], then the angles are complementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are complementary, then [latex]m\angle{A}+m\angle{B}=90^\circ[/latex].

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

## Use a Problem Solving Strategy for Geometry Applications.

- Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
- Identify what you are looking for.
- Name what you are looking for and choose a variable to represent it.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

An angle measures [latex]\text{40}^ \circ[/latex].

1. Find its supplement

2. Find its complement

Write the appropriate formula for the situation and substitute in the given information. [latex]m\angle A+m\angle B=90[/latex] Step 5. Solve the equation. [latex]c+40=90[/latex]

[latex]c=50[/latex] Step 6. Check:

[latex]50+40\stackrel{?}{=}90[/latex]

In the following video we show more examples of how to find the supplement and complement of an angle.

Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[/latex] and [latex]180[/latex] are in numerical order?

Two angles are supplementary. The larger angle is [latex]\text{30}^ \circ[/latex] more than the smaller angle. Find the measure of both angles.

- Question ID 146497, 146496, 146495. Authored by : Lumen Learning. License : CC BY: Attribution
- Determine the Complement and Supplement of a Given Angle. Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/ZQ_L3yJOfqM . License : CC BY: Attribution
- Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

## Missing Angles Practice Questions

Click here for questions, click here for answers.

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## 7.1.5: Using Equations to Solve for Unknown Angles

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Let's figure out missing angles using equations.

Exercise \(\PageIndex{1}\): Is this Enough?

Tyler thinks that this figure has enough information to figure out the values of \(a\) and \(b\).

Do you agree? Explain your reasoning.

Exercise \(\PageIndex{2}\): What Does It Look Like?

Elena and Diego each wrote equations to represent these diagrams. For each diagram, decide which equation you agree with, and solve it. You can assume that angles that look like right angles are indeed right angles.

1. Elena: \(x=35\)

Diego: \(x+35=180\)

2. Elena: \(35+w+41=180\)

Diego: \(w+35=180\)

3. Elena: \(w+35=90\)

Diego: \(2w+35=90\)

4. Elena: \(2w+35=90\)

Diego: \(w+35=90\)

5. Elena: \(w+148=180\)

Diego: \(x+90=148\)

Exercise \(\PageIndex{3}\): Calculate the Measure

Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.

Lines \(l\) and \(m\) are perpendicular.

Are you ready for more?

The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of \(a+b+c\).

- Use a protractor to measure the three angles. Use your measurements to conjecture about the value of \(a+b+c\).
- Find the exact value of \(a+b+c\) by reasoning about the diagram.

To find an unknown angle measure, sometimes it is helpful to write and solve an equation that represents the situation. For example, suppose we want to know the value of \(x\) in this diagram.

Using what we know about vertical angles, we can write the equation \(3x+90=144\) to represent this situation. Then we can solve the equation.

\(\begin{aligned} 3x+90&=144 \\ 3x+90-90&=144-90 \\ 3x&=54 \\ 3x\cdot\frac{1}{3}&=54\cdot\frac{1}{3} \\ x&=18\end{aligned}\)

## Glossary Entries

Definition: Adjacent Angles

Adjacent angles share a side and a vertex.

In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).

Definition: Complementary

Complementary angles have measures that add up to 90 degrees.

For example, a \(15^{\circ}\) angle and a \(75^{\circ}\) angle are complementary.

Definition: Right Angle

A right angle is half of a straight angle. It measures 90 degrees.

Definition: Straight Angle

A straight angle is an angle that forms a straight line. It measures 180 degrees.

Definition: Supplementary

Supplementary angles have measures that add up to 180 degrees.

For example, a \(15^{\circ}\) angle and a \(165^{\circ}\) angle are supplementary.

Definition: Vertical Angles

Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.

For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^{\circ}\), then angle \(DEB\) must also measure \(120^{\circ}\).

Angles \(AED\) and \(BEC\) are another pair of vertical angles.

Exercise \(\PageIndex{4}\)

Segments \(AB\), \(DC\), and \(EC\) intersect at point \(C\). Angle \(DCE\) measures \(148^{\circ}\). Find the value of \(x\).

Exercise \(\PageIndex{5}\)

Line \(l\) is perpendicular to line \(m\). Find the value of \(x\) and \(w\).

Exercise \(\PageIndex{6}\)

If you knew that two angles were complementary and were given the measure of one of those angles, would you be able to find the measure of the other angle? Explain your reasoning.

Exercise \(\PageIndex{7}\)

For each inequality, decide whether the solution is represented by \(x<4.5\) or \(x>4.5\).

- \(-24>-6(x-0.5)\)
- \(-8x+6>-30\)
- \(-2(x+3.2)<-15.4\)

(From Unit 6.3.3)

Exercise \(\PageIndex{8}\)

A runner ran \(\frac{2}{3}\) of a 5 kilometer race in 21 minutes. They ran the entire race at a constant speed.

- How long did it take to run the entire race?
- How many minutes did it take to run 1 kilometer?

(From Unit 4.1.2)

Exercise \(\PageIndex{9}\)

Jada, Elena, and Lin walked a total of 37 miles last week. Jada walked 4 more miles than Elena, and Lin walked 2 more miles than Jada. The diagram represents this situation:

Find the number of miles that they each walked. Explain or show your reasoning.

(From Unit 6.2.6)

Exercise \(\PageIndex{10}\)

Select all the expressions that are equivalent to \(-36x+54y-90\).

- \(-9(4x-6y-10)\)
- \(-18(2x-3y+5)\)
- \(-6(6x+9y-15)\)
- \(18(-2x+3y-5)\)
- \(-2(18x-27y+45)\)
- \(2(-18x+54y-90)\)

(From Unit 6.4.2)

- Maths Questions

## Angles Questions

Angles questions and answers are available here to help students understand how to solve basic angles problems. These questions cover the different types of angles and their measures, and finding the missing angles when a pair or more than two angles are given in a specific relation. You will also get some extra practice questions at the end of the page. These will help you to improve your geometry skills and get a clear understanding of angles.

What are angles?

In geometry, angles are the figures formed by two rays that are connected by a common point called the vertex. We can measure the angles between two lines, rays or line segments using one of the geometric tools called a protractor. Based on the measure of these angles, we can classify them.

The different types of angles are listed below:

- Acute angle (< 90°)
- Obtuse angle (> 90° and < 180°)
- Right angle (= 90°)
- Straight angle (= 180°)
- Reflex angle (> 180° and < 360°)
- Full rotation angle (= 360°)

Also, check: Angles

## Angles Questions and Answers

1. Classify the following angles:

55° < 90°

Thus, 55° is an acute angle.

90° < 146° < 180°

So, 146° is an obtuse angle.

90° is a right angle.

180° < 250° < 360°

Thus, 250° is a reflex angle.

2. Write two examples of obtuse angles and reflex angles.

As we know, obtuse angles are the angles that measure less than 180° and greater than 90°.

Examples: 112°, 177°

Reflex angles measure less than 360° and greater than 180°.

Examples: 210°, 300°

3. Find the measure of an angle which is complementary to 33°.

If the sum of two angles is 90°, they are called complementary angles.

Let x be the angle which is complementary to 33°.

So, x + 33° = 90°

x = 90° – 33° = 57°

Therefore, the required angle is 57°.

4. What is the measure of an angle that is supplementary to 137°?

If the sum of two angles is 180°, they are called supplementary angles.

Let x be the angle which is supplementary to 137°.

So, x + 137° = 180°

x = 180° – 137° = 43°

Hence, the required angle is 43°.

5. If three angles 2x, 3x, and x together form a straight angle, find the angles. Solution:

We know that straight angle = 180°

Given that the angles 2x, 3x, and x form a straight angle.

That means 2x + 3x + x = 180°

2x = 2 × 30° = 60°

3x = 3 × 30° = 90°

Therefore, the angles are 60°, 90° and 30°.

6. Are 125° and 65° supplementary angles?

As we know, the condition for supplementary angles is that they add up to 180°.

Given angles: 125°, 65°

Sum = 125° + 65° = 190°

Thus, 125° and 65° are not supplementary angles.

7. What is the measure of a complete angle?

The measure of a complete angle is 360°.

Two straight angles form a complete angle, i.e., 180° + 180° = 360°.

Four right angles form a complete angle, i.e., 90° + 90° + 90° + 90° = 360°.

8. Find the value of y if (4y + 22)° and (8y – 10)° form a linear pair.

According to the given,

(4y + 22)° + (8y – 10)° = 180°

4y + 22° + 8y – 10° = 180°

12y + 12° = 180°

12y = 180° – 12°

y = 168°/12

9. Three angles at a point are 135°, 75° and x. Find the value of x.

Given angles at a point are: 135°, 75°, x

As we know, the sum of angles at a point = 360°

So, 135° + 75° + x = 360°

210° + x = 360°

x = 360° – 210° = 150°

Therefore, the value of x is 150°.

10. If 3x + 24° and 5x – 16° are congruent, then find the value of x.

Congruent angles mean equal angles.

So, 3x + 24° = 5x – 16°

⇒ 5x – 3x = 24° + 16°

⇒ x = 40°/2

Thus, the value of x is 20°.

## Practice Problems on Angles

- Are 42° and 58° complementary angles?
- How do you find the measure of an angle which is supplementary to 92°?
- What is the condition for a reflex angle?
- If 38° and 2x + 26° form a right angle, find the value of x.
- Classify the following angles: (i) 72°

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## Angle relationships and unknown angle problems

This lesson first explains supplementary and vertical angles. Then we look at a variety of "angle puzzles", or problems with an unknown angle, and solve those using these basic principles. I also write an equation for each problem, to help students learn to both write and solve equations, since this lesson belongs to my pre-algebra course .

Angles in a triangle — video lesson

Math Mammoth Geometry 3 — a self-teaching worktext with explanations & exercises (grades 4-5)

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In order to access this I need to be confident with:

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Here we will learn about angles, including angle rules, angles in polygons and angles in parallel lines.

There are also angles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are angles?

Angles measure the amount of turn required to change direction. At GCSE we can measure angles using a protractor using degrees . If the diagram is not drawn to scale we can determine missing angles by using angle facts (also referred to as angle properties or angle rules ).

There can be multiple different approaches to find a missing angle.

## Angle types

There are different types of angles.

Step-by-step guide: Types of angles

- Angle rules

We can use angle rules to work out missing angles.

Angle rules are facts that we can apply to calculate missing angles in a diagram.

The five key angle facts that are used widely within the topic are:

- Angles on a straight line

The sum of angles on a straight line is always equal to \bf{180^{o}.}

A straight line would be considered to be half of a full turn; if you were standing on the line facing towards one end, you would have to turn 180 degrees to face the other end of the line.

A straight line can be called a straight angle if there is a vertex on the line and the turn around that vertex is 180^{o}.

- Angles at a point

The sum of angles at a point is always equal to \bf{360^{o}} .

A point would be considered to be a full turn; if you were standing at the point facing in one direction, you would have to turn 360 degrees to return back to your original position.

- Complementary angles

The sum of complementary angles is always equal to \bf{90^{o}} .

Complementary angles therefore make up a right angle.

These angles do not need to be together and form a right angle. If any two angles sum to 90^o they are complementary.

- Supplementary angles

The sum of supplementary angles is always equal to \bf{180^{o}} .

Supplementary angles therefore make up a straight line.

These angles do not need to be together on a straight line. If any two angles sum to 180^o they are supplementary.

- Vertically opposite angles

Vertically opposite angles are equal .

This occurs when two straight lines meet (intersect) at a point known as a vertex , forming an x shape where the opposite pairs of angles are the same size. Also, two adjacent angles are supplementary (they add to equal 180^o ).

Step-by-step guide: Angle rules

- Angles in polygons

We can calculate the interior and exterior angles of any polygon.

- Interior angles

The sum of interior angles in any n -sided shape is determined using the formula,

One interior angle of a regular polygon with n -sides is determined using the formula,

For an irregular polygon, the missing angle is calculated by subtracting all of the known angles from the total sum of the interior angles of the polygon.

- Exterior angles

The sum of exterior angles for any polygon is \bf{360^{o}} .

Whereas the interior angle sum is different for each n -sided shape, the exterior angle sum is always 360^{o}, regardless of how many sides the polygon has.

This is because, as you walk around the perimeter of the shape, the exterior angle is the turn from the direction of one edge to the next edge of the polygon.

For a regular polygon, each exterior angle is equal to 360 divided by the number of sides, n, and so

For an irregular polygon, the unknown exterior angle is calculated by subtracting the known exterior angles from 360^{o}.

The sum of an exterior angle and its adjacent interior angle is 180^{circ} , because they both lie on a straight line.

- Angles in a triangle

The sum of angles in a triangle is \bf{180^{o}} .

Remember that there are four different types of triangles, each with a specific angle property.

- Angles in a quadrilateral

The sum of angles in a quadrilateral is \bf{360^{o}} .

Any quadrilateral can be constructed from two adjacent triangles. This means that as the angle sum of a triangle is equal to 180^{o}, two triangles would have an angle sum of 360^{o}.

There are several different types of quadrilaterals, each with a specific angle property.

Step-by-step guide: Angle in polygons

- Angles in parallel lines

Angles in parallel lines are facts that can be applied to calculate missing angles within a pair of parallel lines. The three key angle facts that are used when looking at angles in parallel lines are,

- Corresponding angles
- Alternate angles
- Co-interior angles

Corresponding angles are equal .

When we intersect a pair of parallel lines with a transversal (another straight line), corresponding angles are the angles that occur on the same side of the transversal line. They are either both obtuse or both acute.

Alternate angles are equal .

When we intersect a pair of parallel lines with a transversal, alternate angles occur on opposite sides of the transversal line. They are either both obtuse or both acute.

The sum of two co-interior angles is \bf{180^{o}} .

Co-interior angles occur in between two parallel lines when they are intersected by a transversal . The two angles that occur on the same side of the transversal always add up to 180^{o}.

Step-by-step guide: Angles in parallel lines

## How to use angles

We can use angles in lots of different contexts.

We will learn about,

- Types of angles

## Explain how to use angles

## Angles worksheet

Get your free angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

## Angle rules examples

Example 1: angles on a straight line.

Calculate the value of x.

Add all known angles.

The angle highlighted as a square is a right angle. A right angle measures 90^{o}. Adding the angles together, we can form the expression

2 Subtract the angle sum from \bf{180^{o}} .

As the sum of angles on a straight line total 180^{o},

3 Form and solve the equation.

Here we have no equation to solve as the missing angle is 20^{o}.

## Example 2: angles at a point

Subtract the angle sum from \bf{360^{o}} .

Form and solve the equation.

Here there is no equation to solve as we know the angle x=60^{o}.

## Example 3: complementary angles

Angle AOB is complementary to BOC. Determine the size of the angle AOB.

Identify which angles are complementary .

Here, the angles AOB and BOC are complementary.

Clearly identify which of the unknown angles the question is asking you to find the value of.

The question would like us to calculate the angle AOB.

Solve the problem and give reasons where applicable .

As the sum of the two angles is 90 degrees, forming an equation, we have

x+10+3x=90,

Solving this for x, we have

\begin{aligned} 4x&=90-10\\\\ 4x&=80\\\\ x&=20. \end{aligned}

Clearly state the answer using angle terminology.

We need to determine the size of angle AOB for the solution and so we need to substitute x=20 into x+10.

Angle AOB = x+10=20+10=30^{o}.

## Example 4: supplementary angles

Given that AC is a straight line, determine the size of angle AOB.

Identify which angles are supplementary .

As the line AC is a straight line, the two angles AOB and BOC are supplementary. This means that we can use the angle rule, “the sum of supplementary angles is 180^{o} ”.

We need to determine the size of angle AOB.

Solve the problem and give reasons where applicable.

Adding the two angles 5x+35 and x+25 is equal to 180^{o}, which gives us the equation

5x+35+x+25=180.

Simplifying the left side of the equation, we have

Subtracting 60 from both sides of the equation, we have

6x=180-60=120.

Dividing both sides by 6, we have

x=120\div{6}=20.

We need to determine the size of angle AOB for the solution and so we need to substitute x=20 into angle AOB \ (5x+35).

Angle AOB = 5x+35=(5\times{20})+35=135^{o}.

## Example 5: vertically opposite angles

Given that AC and BD are straight intersecting lines at the point O, determine the size of angle COD.

Identify which angles are vertically opposite to one another.

Angle COD is vertically opposite angle AOB and so angle COD = angle AOB.

Angle AOD is vertically opposite angle BOC and so angle AOD = angle BOC.

We need to calculate the size of angle COD \ (2x).

We need to inspect the lines and angles in the diagram to see what other rule(s) we need to apply.

As BD is a straight line, the two angles of 72^o and 2x are supplementary.

This means that if we use the rule “the sum of supplementary angles is 180^{o} ,” we can calculate the size of the angle 2x and hence the missing angle COD.

Forming an equation, we have

Subtracting 72 from both sides of the equation, and then dividing both sides by 2, we have

2x=180-72=108,

x=108\div{2}=54.

As x=54, \ 2x=2\times{x}=2\times{54}=108. As vertically opposite angles are equal,

Angle COD = 108^{o}.

## Example 6: interior angles of a regular pentagon

What is the interior angle sum of a regular pentagon?

Identify how many sides the polygon has.

A pentagon has 5 sides.

Identify if the polygon is regular or irregular.

The polygon is regular.

If possible work out how many triangles could be created within the polygon by drawing lines from one vertex to all other vertices.

A regular pentagon contains 3 triangles.

Multiply the number of triangles by \bf{180} to calculate the sum of the interior angles.

State your findings.

The interior angle sum of a regular pentagon is 540^{o}.

## Example 7: angles in a triangle

What type of triangle is ABC?

Add up the other angles within the triangle.

Subtract this total from \bf{180^{o}} .

The remaining angle is 90 degrees and so this is a right angle triangle.

## Example 8: angles in a quadrilateral

ABCD is a kite. Calculate the size of angle ADO.

Use angle properties to determine any interior angles.

Angle ABC and ADC are equal as a kite has one pair of equal angles.

This means that ABC = ADC = 2x+10.

Updating the diagram with this information, we have

Add all known interior angles.

The remaining angles must therefore total 260^{o} and so we can form the equation

2x+10+2x+10=260.

Simplifying the left hand side of the equation, we have

Subtract 20 from both sides,

4x=260-20=240.

Divide both sides by 4,

x=240\div{4}=60.

ADO = 60^{o}.

## Example 9: exterior angles

Calculate the exterior angle x for the hexagon below.

Identify the number of sides in any polygon given in the question.

This irregular hexagon has 6 sides.

Identify what the question is asking.

We need to find the value of the exterior angle, x.

Solve the problem using the information you have already gathered.

The known exterior angles of this hexagon are: 70^{o}, \ 50^{o}, \ 20^{o}, and 90^{o} (the right angle).

As the interior angle is supplementary to the exterior angle at F, the exterior angle at F is equal to 180-100 = 80^{o}.

We now have the known angles: 70^{o}, \ 50^{o}, \ 20^{o}, \ 90^{o}, \ 80^{o} and the unknown angle x.

Subtracting the known angles from 360, we have

x=360-(70+50+20+90+80)=360-310=50.

The missing exterior angle x=50^{o}.

## Example 10: alternate angles

Calculate the size of the angle x.

Highlight the angle(s) that you already know.

We know angle BGH and we want to find angle GHC.

State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.

BGH is alternate to angle GHC (our unknown value x ), so we can state that x=72^{o}.

Use basic angle facts to calculate the missing angle.

This step isn’t required as the solution has been calculated.

## Example 11: corresponding angles

Calculate the value of y.

We do not know the numerical size of any angle in the diagram however we can use angles BIG, \ DJI and EKJ to determine the value of y.

The angle GIB is corresponding to GKF. As EF is a straight line we can calculate the value for x, which will help us to calculate the value of y.

As GIB is corresponding to GKF, angle GKF = 7x.

As EF is a straight line and the sum of angles on a straight line is 180, we can form the equation 2x+7x=180. Solving this equation we have

\begin{aligned} 9x&=180\\\\ x&=180\div{9}\\\\ x&=20 \end{aligned} .

As GKF = 7x, \ 7 \times 20=140 and so GKF = 140^{o}.

State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.

Angle GKF is corresponding to GJD and so as corresponding angles are equal, GJD = 140^o and so

## Example 12: co-interior angles

We know the two angles BGH and GHD.

BGH is co-interior to GHD and so their angle sum is 180^{o}. Forming an equation for the two angles, we have

2x+50+2x+10=180.

Next, we subtract 60 from both sides

4x=180-60=120.

And finally divide by 4.

x=120\div{4}=30

Repeat this process until the required missing angle is calculated.

## Common misconceptions

- Measuring angles with a protractor when the diagrams are not drawn accurately

When the question states that the diagram is not drawn accurately we cannot simply measure the missing angles using a protractor. We need to use angle facts to calculate the missing angle.

- Angles on a straight line and vertically opposite angle rules

With the straight line rule that only adjacent angles are considered, and for vertically opposite angles the lines must be straight.

- Exterior angles of a polygon

Exterior angles of a polygon have to travel in the same direction for the sum to be 360^{o}.

- Angles in triangles

Pairing up the incorrect angles in an isosceles triangle and using exterior angles when calculating the interior angle sum of a triangle.

## Practice angles questions

1. Calculate the value of x.

The sum of angles on a straight line is 180^{o}.

2. Calculate the value of x.

The sum of angles at a point is 360^{o}.

3. AOC is a right angle. Calculate the value of angle BOC.

The sum of two complementary angles is 90^{o}.

4. AB is a straight line. Calculate the size of angle AOC.

As AB is a straight line, angle AOC is supplementary to angle BOC.

Angle AOC = 3x+x=4x and so

5. The two straight lines AC and BD intersect at the point O. Calculate the value of x.

As AC and BD are straight lines, angle COD is vertically opposite angle AOB.

As vertically opposite angles are equal, we can form the equation

6. The hexagon below has two lines of symmetry. Calculate the size of the interior angle at A.

The interior angle at C = 2x given the vertical line of symmetry.

The interior angle at D,E, and F can be found using the horizontal line of symmetry: D = 2x, \ E = 150^{o}, and F = 2x.

The sum of angles in the hexagon is therefore

As the interior angle at A is equal to 2x,

7. Calculate the value of y.

As the sum of angles in a triangle is 180^{o},

8. Calculate the interior angle at A.

The sum of angles in a quadrilateral is 360^{o}.

9. Calculate the interior angle of a regular octagon.

10. Which diagram only needs us to use alternate angles to determine the value of x?

BJH is alternate to JKL.

JKL is alternate to KLF.

11. Which diagram needs us to use corresponding angles to determine the value of x?

BJH is corresponding to KJF.

The sum of angles in a triangle is 180^o and so angle

Angle JKL is corresponding to HIK.

12. Which diagram uses co-interior angles to calculate the value of x?

DJI is co-interior to BIJ.

EF is a straight line. The sum of angles on a straight line is 180^{o}.

## Angles GCSE questions

1. A sock design requires four colours of thread. The pie chart below shows the proportion of each colour.

The ratio of white to yellow is 2 : 1. What fraction of the thread is white?

2. ABC and ACD are two congruent isosceles triangles.

Show that the angle BCD is double the angle BAD.

State any angle rules you use.

BAC = ABC = 20^o and base angles in an isosceles triangle are equal.

ACB = 180-(20 + 20) = 140^o and the sum of angles in a triangle is 180^{o}.

ACD = ACB = 140^o and the two triangles are congruent.

BAD = 20 + 20 = 40^o and the two triangles are congruent.

BCD = 360-(140 + 140) = 80^o and the sum of angles at a point is 360^{o}.

3. Below are the three lines AB, \ CD, and EF, intersected by the line GH.

Show that the lines AB, \ CD, and EF are not parallel.

If AB and CD are parallel, 4x+2x+5=180.

DJK is corresponding to BIJ.

The sum of angles on a straight line total 180^{o}.

x must be the same for both equations.

Alternative Method

If AB and EF are parallel, 4x+3x-15=180.

FKH is corresponding to BIJ.

## Learning checklist

You have now learned how to:

- Recognise angles as a property of shape or a description of a turn
- Apply the properties of angles at a point, angles at a point on a straight line, and vertically opposite angles
- Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons
- Distinguish between regular and irregular polygons based on reasoning about equal sides and angles
- Understand and use the relationship between parallel lines and alternate and corresponding angles

## The next lessons are

- Pythagoras theorem

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## Solving Triangles

"Solving" means finding missing sides and angles.

## Six Different Types

If you need to solve a triangle right now choose one of the six options below:

Which Sides or Angles do you know already? (Click on the image or link)

... or read on to find out how you can become an expert triangle solver :

## Your Solving Toolbox

Want to learn to solve triangles?

Imagine you are " The Solver " ... ... the one they ask for when a triangle needs solving!

In your solving toolbox (along with your pen, paper and calculator) you have these 3 equations:

## 1. Angles Add to 180° :

A + B + C = 180°

When you know two angles you can find the third.

## 2. Law of Sines (the Sine Rule):

When there is an angle opposite a side, this equation comes to the rescue.

Note: angle A is opposite side a, B is opposite b, and C is opposite c.

## 3. Law of Cosines (the Cosine Rule):

This is the hardest to use (and remember) but it is sometimes needed to get you out of difficult situations.

It is an enhanced version of the Pythagoras Theorem that works on any triangle.

## Six Different Types (More Detail)

There are SIX different types of puzzles you may need to solve. Get familiar with them:

This means we are given all three angles of a triangle, but no sides.

AAA triangles are impossible to solve further since there are is nothing to show us size ... we know the shape but not how big it is.

We need to know at least one side to go further. See Solving "AAA" Triangles .

This mean we are given two angles of a triangle and one side, which is not the side adjacent to the two given angles.

Such a triangle can be solved by using Angles of a Triangle to find the other angle, and The Law of Sines to find each of the other two sides. See Solving "AAS" Triangles .

This means we are given two angles of a triangle and one side, which is the side adjacent to the two given angles.

In this case we find the third angle by using Angles of a Triangle , then use The Law of Sines to find each of the other two sides. See Solving "ASA" Triangles .

This means we are given two sides and the included angle.

For this type of triangle, we must use The Law of Cosines first to calculate the third side of the triangle; then we can use The Law of Sines to find one of the other two angles, and finally use Angles of a Triangle to find the last angle. See Solving "SAS" Triangles .

This means we are given two sides and one angle that is not the included angle.

In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side. See Solving "SSA" Triangles .

This means we are given all three sides of a triangle, but no angles.

In this case, we have no choice. We must use The Law of Cosines first to find any one of the three angles, then we can use The Law of Sines (or use The Law of Cosines again) to find a second angle, and finally Angles of a Triangle to find the third angle. See Solving "SSS" Triangles .

## Tips to Solving

Here is some simple advice:

When the triangle has a right angle, then use it, that is usually much simpler.

When two angles are known, work out the third using Angles of a Triangle Add to 180° .

Try The Law of Sines before the The Law of Cosines as it is easier to use.

## Missing Angles

Related worksheets.

Find the missing angles in a triangle, around a point, in a quadrilateral, find opposite and supplementary angles, or find angles which require multi-step problem solving skills.

For more shape and space resources click here.

## Game Objectives

New Maths Curriculum:

Year 3: Identify right angles, recognise that two right angles make a half-turn, three make three quarters of a turn and four a complete turn; identify whether angles are greater than or less than a right angle

Year 4: Identify acute and obtuse angles and compare and order angles up to two right angles by size

Year 5: Identify: multiples of 90°, angles at a point on a straight line and ½ a turn (total 180°); angles at a point and one whole turn (total 360°); reflex angles; and compare different angles

Year 6: Find unknown angles where they meet at a point, are on a straight line, and are vertically opposite

## Angle Word Problems

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## Find and Identify Angles in the Real World

## How are Angles Used in the Real World?

Angles are everywhere! It’s great to have an angle hunt or to search for angles in the real world because students and teachers start to see angles that they may not have noticed before. When you look for angles in objects, you can find patterns and see how angles work together.

I love helping students find and identify angles in real life objects because they start to see how one angle might have another angle that goes with it. One time I had a drawing of a house and asked students to find one right angle. I was thinking that they would point out the right angle inside the house, but I had some students find the right angle of the outside, where the house meets the ground, or a right angle inside the door or window. Just this simple perspective of finding the same angle in different places helps reveal more views of angles and of the house. This helps students see how angles are made and even introduces concepts like supplementary and complementary angles.

As students discover angles in real life objects, they also learn about different types of angles and how angles help to make up shapes. I’ve had students search for and find types of angles in everyday objects and share what they found. If I can get my hands on a camera, then I have students take pictures of objects, print them out and outline and identify angles inside their photos.

## Set up for the Find and Identify Angles Lesson

- Set up themed stations around the room with photographs of items from those places (photographs are provided on pages 2-11 of the Find and Identify Angles in the Real World worksheet , but you can add your own, especially if you have photographs of local spots that students would recognize): Aquarium, At the Park, Amusement Park, In the City, At the Beach.If possible, print the photographs in color for each station. You can also put the photographs into plastic sleeves. If you use page protectors, have thin tipped dry erase markers available.
- Provide each student with the Find and Identify Angles in the Real World worksheet (these can be printed in black and white).
- Provide students with colored pencils or thin tipped markers to record the angles onto the pictures on their worksheet.
- If you have a projector, set it up to show the image of a playground (shown below or a similar one) to project the image and then be able to draw images over the projection on the board, remove the image and see the angles on their own.
- Provide students with blank paper and rulers, and, if available, protractors. (Students do not need to use the protractors to measure the angles, but this is a good time to introduce the protractor and how it helps students identify acute, obtuse, and right angles.)

## Launch the Activity: Angles in the Real World

Project a photograph of a playground on a whiteboard or large piece of paper. The students have the same photograph on page 1 of their Find and Identify Angles worksheet and should follow along with as you trace angles onto the photograph, they can do so on their worksheet. If you want students to be able to practice before drawing onto the worksheets, you can provide plastic sleeves and students can use dry erase markers.

## Here is an image that provides examples of each type of angle:

With the image projected, you can start to name and identify types of angles. Start by asking your students if they see angles. Then challenge students to see how many angles they can find. As students find angles, take a marker and highlight at least one angle of each type (acute, right, obtuse, and straight). Draw the marker lines over the angles in the image and identify each one, such as shown below.

## Here is an example of an angle in this image:

Ask students if they can identify or name this type of angle. It is an acute angle .

Then show an acute angle outside of the picture. You can do this by turning off the projector and showing the angle by itself. Like this:

Have the students sketch an angle similar to this on their worksheet and identify it as an acute angle .

Now help students to identify an obtuse angle , a right angle , and a straight angle in this image, reviewing the definition of each as you find it. You can identify and show these angles yourself, or ask for volunteers to come up and find an angle in the photo. Students can find these angles in their own image on the worksheet too.

## Make sure every student fills out an example and definition for each type of angle:

## Here is an example of the image with each type of angle.

You can color code if you use plastic sleeves, but students should sketch angles with pencil first on their worksheet and then color code.

After finding angles on the playground, and naming angles on the worksheet, divide the students into groups and have them visit each themed station. Students should use the color photographs at each station to find and identify angles. They should find the corresponding photograph on their worksheet, which will give them instructions for which types of angles to find. Then they will draw those angles (preferably in color) onto the image on their worksheet.

Each station should have multiple photographs so students have the option to work independently, even if they are grouped at a station.

Photographs for each station are located on pages 2-11 of the Find and Identify Angles worksheet , but you can also include more of your own, especially if you have photographs of local places that students would recognize and that correspond to the themes below. The instructions for the types of angles students should search for in each photograph are located beneath each photograph on the student worksheets.

Below are the station themes and lists of objects that would likely have angles to identify, in case you want to supplement with local pictures:

- Aquarium: starfish, shark fin, coral, plants, fish tail, fish head, crab claws, lobster, turtle, clam shell, scuba diver, flippers, submarine
- At the park : Slide, swings, seesaw, baseball diamond, soccer net, basketball court, jungle gym, trees, bikes, strollers
- Amusement Park: Swings, roller coaster, arcade, games, stroller, ice cream cones, snow cones, paths, train, gondola, pizza, slides
- In the city: Skyscrapers, angled buildings, sidewalks, subway, tracks, bikes, strollers, doors, windows, food carts, store windows, taxi cabs, crosswalks, cones, parking spots
- At the beach : Umbrellas, towels, buckets, shovels, sand castles, sand piles, boogie boards, sail boat, paddle board, kites, coolers, picnic baskets, bathing suits, shells, rocks

Set a timer for every 8-10 minutes, depending on the length of your class period, and have students rotate to a new station. Students do not need to do every photograph and do not need to work in the order the photographs appear on the worksheet, but they should keep working and tracing angles until the time is up at that station.

While students are working at the stations, circulate through the room and ask questions, encouraging students to find new angles they have not yet noticed.

## Angles in the Real World Activity Reflection

Lead students in small or whole group discussions around reflective questions, such as the following:

- Did you find more acute or more obtuse angles?
- Are there more right angles than other kinds of angles?
- Did you find one type of angle and then see another one that goes alongside it?
- Did each station get easier or harder?
- Which station did you find the most challenging?

There are also reflection questions on the worksheet that students can complete on their own or in groups.

## Angles in the Real World Extensions

- Students can go around school, at home, or in the community to take pictures with a tablet or camera and draw and sketch angles on the photos. Students could also add their own photographs (with angles sketched on) to the existing themed stations.
- Students can create a poster or slides presentation that shows images that have angles. Maybe one image highlights only acute angles, another shows only obtuse and another shows only right angles.
- Create a “Seek and Find” book or poster where there are “hidden” angles and include an answer key.

## FREE Find and Identify Angles in Real Life Objects Worksheets and Resources

These are all PDF Files. They will open and print easily. The Student Edition Files are labeled SE and the Teacher Editions Files are labeled TE. Click the links below to download the different resources.

- 7-2 Assignment SE – Angles
- 7-2 Assignment TE – Angles ( Member Only )
- 7-2 Bell Work SE – Angles
- 7-2 Bell Work TE – Angles ( Member Only )
- 7-2 Exit Quiz SE – Angles
- 7-2 Exit Quiz TE – Angles ( Member Only )
- 7-2 Guided Notes SE – Angles
- 7-2 Guided Notes TE – Angles ( Member Only )
- 7-2 Interactive Notebook SE – Angles
- 7-2 Lesson Plan – Angles
- 7-2 Online Activities – Angles
- 7-2 Slide Show – Angles

## Find and Identify Angles in Real Life Objects Worksheets and Resources

To get the Editable versions of these files Join us inside the Math Teacher Coach Community! This is where we keep our full curriculum of 4th Grade Math Lessons and Activities.

- 7-2 Assignment SE – Angles ( Member Only )
- 7-2 Bell Work SE – Angles ( Member Only )
- 7-2 Exit Quiz SE – Angles ( Member Only )
- 7-2 Guided Notes SE – Angles ( Member Only )
- 7-2 Interactive Notebook SE – Angles ( Member Only )
- 7-2 Lesson Plan – Angles ( Member Only )
- 7-2 Online Activities – Angles ( Member Only )
- 7-2 Slide Show – Angles ( Member Only )

## Do You Want 4th Grade Math Activities Delivered To Your Inbox Everyday?

If you would like our 4th Grade Math, 5th Grade Math, 6th Grade Math, 7th Grade Math, and 8th Grade Math Resources Emailed to you Daily Click Here .

## Don’t Forget to Pin this lesson on Find and Identify Angles in Real Life Objects…

## Want to see the rest of the activities for Unit 7– Geometry ?

- 7-1 The Undefined Terms in Geometry
- 7-3 Parallel and Perpendicular Lines
- 7-4 Measuring and Sketching Angles
- 7-5 Addition of Angle Measures

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- Master Geometry Formulas: A Comprehensive Guide for Academic Success

## Geometry Formulas Unraveled: Your Go-To Resource for Assignments

Geometry, a captivating branch of mathematics, ventures into the nuanced exploration of shapes, sizes, and dimensions within the space we occupy. The intricate world of geometry presents students with a labyrinth of concepts, demanding a profound comprehension of formulas. These formulas are not mere mathematical abstractions; they are the key to unlocking the secrets hidden within angles, polygons, circles, and three-dimensional spaces. Whether engrossed in challenging assignments, gearing up for examinations, or simply striving to grasp the fundamental principles, students stand to gain significantly from a comprehensive resource. This blog serves as a beacon, illuminating the path toward mastery in this mathematical domain by providing practical insights, real-world applications, and a holistic approach that transcends rote memorization. For those seeking assistance with your Geometry assignment , this guide aims to be an invaluable tool in achieving success and mastering the fascinating world of geometry.

As we conclude this exploration, it becomes evident that geometry is not just a theoretical construct but a dynamic tool. Armed with a profound understanding of these formulas, students can navigate the intricate terrain of mathematical problem-solving with confidence. Whether unraveling the mysteries of angles, exploring the symmetry of shapes, or delving into the realms of trigonometry, this blog encapsulates the essence of geometry, offering a comprehensive resource that transcends the boundaries of a traditional academic guide. In the tapestry of mathematical knowledge, geometry unfolds as a vibrant thread, weaving together the abstract and the concrete, and in this journey, we hope to be your steadfast companion, illuminating the path to geometric proficiency in your academic endeavors.

## The Basics of Geometry

In laying the groundwork for understanding the intricate realm of geometry, it's imperative to revisit the fundamental concepts that underpin this mathematical discipline. At its core, geometry explores the properties and relationships of various elements in the space we inhabit. Beginning with the basic building blocks such as points, lines, and angles, students gain a foundational understanding that serves as a prerequisite for more advanced geometric principles. Euclidean geometry, with its axioms and postulates, forms the bedrock upon which many geometric theorems rest. Delving into the terminology and classifications of shapes, including triangles, quadrilaterals, and polygons, provides the scaffolding for the application of formulas governing perimeter, area, and other characteristics of these fundamental geometric entities. This section serves as an essential refresher, emphasizing the importance of mastering these basic concepts before venturing into the more complex and nuanced realms of geometric reasoning. As students grasp the basics, they not only develop the necessary skills for problem-solving but also cultivate a solid foundation upon which the edifice of geometric knowledge can be constructed, paving the way for a deeper exploration of the geometrical intricacies that lie ahead.

## Properties of Shapes

We delve into the multifaceted realm of shape properties, unraveling the fundamental formulas that underpin geometric understanding. From the simplicity of triangles to the intricacies of polygons, each shape brings a unique set of properties and challenges. Exploring the concept of perimeter, we uncover the formulaic essentials for calculating the total length of a shape's boundary, shedding light on its significance in practical applications. Moving on to the realm of area, we navigate the intricacies of surface measurement, illustrating how these formulas are not just mathematical abstractions but crucial tools for quantifying space within shapes. Through a comprehensive examination of shapes like rectangles, circles, and irregular polygons, we decode the formulas for area, shedding light on the interplay between shape dimensions and space occupancy. Additionally, we unveil the importance of understanding the Pythagorean theorem in the context of right-angled triangles, highlighting its role in calculating sides and angles. As we unravel the properties of shapes, we emphasize the practical relevance of these formulas, showcasing how they extend beyond the classroom into real-world scenarios, from calculating material requirements in construction to determining land areas in urban planning. This section serves as a foundational guide, empowering students and enthusiasts alike with the tools needed to navigate the dynamic landscape of shape properties and apply them adeptly in problem-solving and analysis.

## Circle Formulas and Applications

In the realm of geometry, circles stand out as ubiquitous and intriguing entities, playing a pivotal role in numerous facets of our daily lives. Circle formulas, encapsulating essential parameters like circumference, area, and sector calculations, serve as the mathematical backbone for understanding and manipulating these geometric wonders. The circumference, representing the boundary of a circle, is determined by the formula C = 2πr, where r is the radius. This seemingly simple formula finds applications in a myriad of fields, from calculating the length of a bicycle tire to determining the orbits of celestial bodies. The area of a circle, expressed as A = πr², elucidates the extent of space enclosed by the circular boundary, proving indispensable in fields such as land surveying and urban planning. Sector calculations, offering insights into portions of a circle, involve formulas like the area of a sector (A_sector = 0.5r²θ) and the arc length (L = rθ), where θ represents the central angle. These formulas find resonance in diverse areas, ranging from pie-chart constructions to the assessment of angles of elevation in trigonometry. As we unravel the applications of circle formulas, their ubiquity becomes apparent, resonating not only in mathematical problem-solving but also in the intricate tapestry of our tangible, real-world experiences. Whether in the precision of scientific measurements or the aesthetics of design, the understanding and application of circle formulas serve as a cornerstone in comprehending the geometrical symphony that surrounds us.

## Angle Relationships and Trigonometry

Angle relationships and trigonometry form a crucial aspect of geometry, offering a deeper understanding of the relationships between angles and their applications in real-world scenarios. In the realm of angle relationships, the study involves exploring the fundamental theorems governing angles, such as the vertical angle theorem, complementary and supplementary angles, and the transversal angle theorems. These principles lay the groundwork for solving intricate geometric problems, providing a systematic approach to angle-related inquiries. Trigonometry, on the other hand, introduces the study of the ratios and functions of angles within right-angled triangles, including sine, cosine, and tangent. These trigonometric functions serve as powerful tools in measuring distances, heights, and angles, extending their applicability to fields ranging from physics and engineering to astronomy. Understanding the relationships between angles and the application of trigonometric functions empowers individuals to solve complex problems involving triangles and circular motion. Moreover, trigonometry serves as a bridge between geometry and algebra, showcasing the interconnectedness of mathematical concepts. As students delve into angle relationships and trigonometry, they not only gain proficiency in handling geometric challenges but also acquire valuable problem-solving skills with broader implications across various disciplines.

## Three-Dimensional Geometry

Three-dimensional geometry adds a layer of complexity to our understanding of space, introducing a dynamic realm of shapes and structures. In this multidimensional space, volumes and surfaces take center stage, challenging us to explore formulas that go beyond the confines of traditional two-dimensional geometry. From calculating the volume of prisms and cylinders to determining the surface areas of spheres and cones, three-dimensional geometry provides a rich tapestry of mathematical concepts. The formulas associated with these shapes offer practical insights into real-world applications, influencing fields ranging from architecture to physics. Understanding spatial relationships becomes crucial as we navigate the intricacies of three-dimensional space, requiring a keen awareness of how shapes interact and transform. As we delve into this dimension, the significance of these formulas becomes evident in their role as problem-solving tools for architects designing structures, engineers conceptualizing complex systems, and scientists modeling the behavior of physical objects. Three-dimensional geometry not only expands our mathematical toolkit but also deepens our appreciation for the spatial dimensions that shape the world around us, fostering a connection between abstract mathematical concepts and their tangible manifestations in our physical reality. In this realm of mathematical exploration, students and professionals alike find themselves unraveling the complexities of volumes, surfaces, and spatial relationships, opening doors to new dimensions of understanding and problem-solving.

## Transformational Geometry

Transformational geometry is a captivating branch that takes geometry beyond static shapes and explores the dynamic world of spatial transformations. In this realm, geometric figures undergo changes in position, size, and orientation, opening up a rich array of mathematical possibilities. Transformations include translations, where figures move parallel to themselves; rotations, where figures pivot around a fixed point; reflections, which involve flipping figures across a line; and dilations, where figures scale up or down. Each transformation has its set of formulas and rules, serving as tools to manipulate and analyze shapes in various contexts. Understanding transformational geometry is not only vital for solving intricate mathematical problems but also has practical applications in fields like computer graphics, art, and design. In computer-aided design (CAD), for instance, transformations play a crucial role in modeling and rendering three-dimensional objects. Furthermore, in art and animations, transformations bring life to characters and scenes by seamlessly altering their appearance. Transformational geometry adds a dynamic layer to the study of shapes, providing a bridge between mathematical abstraction and real-world applications. Mastery of these transformations empowers individuals to explore the fluidity and adaptability inherent in geometric figures, enhancing problem-solving skills and fostering a deeper appreciation for the dynamic nature of our geometric universe.

## Real-World Problem Solving

In the realm of geometry, the application of formulas extends far beyond the confines of a classroom. Section 7 unravels the intricate tapestry of real-world problem-solving, demonstrating the practical implications of geometric concepts in various fields. Architects employ geometry to design structurally sound buildings, ensuring that shapes, angles, and dimensions harmonize for both aesthetic appeal and stability. Engineers rely on geometric principles to calculate stress distributions, optimize designs, and construct efficient structures. The application extends to physics, where geometric formulas come into play when determining distances, trajectories, and spatial relationships between objects in motion. From the intricate calculations involved in satellite orbit trajectories to the design of everyday objects like bridges and vehicles, geometry underpins the fabric of our built environment. Moreover, advancements in technology leverage geometric concepts, enabling innovations in computer graphics, 3D modeling, and virtual simulations. Whether it's the precise mapping of geographical landscapes or the development of cutting-edge virtual reality applications, geometry serves as the silent architect shaping the digital and physical worlds. As students immerse themselves in real-world problem-solving scenarios, they not only reinforce their understanding of geometric principles but also recognize the profound impact these formulas have on shaping the world we inhabit. Thus, Section 7 illuminates the transformative role of geometry, bridging the gap between theoretical knowledge and its tangible manifestations in the complex, multidimensional landscapes of architecture, engineering, physics, and technology.

## Conclusion:

In conclusion, the journey through the intricacies of geometry formulas has revealed the underlying structure and beauty inherent in the mathematical world. From foundational concepts to advanced three-dimensional applications, this comprehensive guide serves as a valuable resource for students grappling with assignments and seeking a deeper understanding of geometric principles. As we've unraveled the mysteries of shapes, angles, circles, and transformations, it's evident that geometry is not merely an abstract discipline but a dynamic tool with practical applications across various fields. The real-world problem-solving section highlights the versatility of geometric formulas in architecture, engineering, and physics, showcasing their relevance beyond the classroom. Armed with this knowledge, students can approach assignments with confidence, recognizing the connections between theoretical concepts and their tangible implications. Moreover, the exploration of transformational geometry underscores the dynamic nature of shapes, paving the way for applications in computer graphics and animation. As we navigate the vast landscape of geometric principles, one can appreciate how these formulas are not just academic exercises but essential tools for understanding and shaping the world around us. With this go-to resource in hand, students are equipped to conquer the challenges of geometry, unveiling the elegance and precision that underlie our geometric reality.

## Post a comment...

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## Mathematics > Metric Geometry

Title: interior angle sums of geodesic triangles and translation-like isoptic surfaces in sol geometry.

Abstract: After having investigated the geodesic triangles and their angle sums in Nil and $Sl\times\mathbb{R}$ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $\pi$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Molnár in \cite{M97}.

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## A new virtual interpolation technology with range as object

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- Published: 17 May 2024

## Cite this article

- Yunxiu Yang 2 ,
- Wendong Chen 3 &
- Qin Shu 1

Virtual interpolation technology can be applied to direction-of-arrival (DOA) estimation as a preprocessing technique to achieve the DOA estimation for any array. In order to solve the angle-sensitive problem in virtual interpolation technology, a range interpolation transformation method is proposed. Taking the range of the angle function as the interpolating object, and modifying the interpolating step according to the mapping between the angle and the range, a high-performance interpolating method with less computation is realized. For large region virtual interpolation, on the other hand, based on multi-region interpolation technology, the region is divided into multiple nested regions to achieve high-performance transformation results and can be used to improve the interpolation accuracy in the process of range interpolation technology. The simulation results show that the range interpolation transform method has obvious advantages over the classical transform method in both transformation error and DOA estimation performance, and the angle-sensitive problem is effectively alleviated. The nested virtual interpolation can be used to achieve high-performance virtual interpolation and improve the accuracy of interpolation in large regions.

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Tao Li & Qin Shu

Southwest Institute of Technical Physics, Chengdu, 610041, China

Yunxiu Yang

Beijing Institute of Remote Sensing Equipment, Beijing, 100084, China

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Li, T., Yang, Y., Chen, W. et al. A new virtual interpolation technology with range as object. SIViP (2024). https://doi.org/10.1007/s11760-024-03151-y

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DOI : https://doi.org/10.1007/s11760-024-03151-y

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## Moms are flocking to use desks that have cribs attached. They say it allows them to balance motherhood with their careers.

- The desks have a play area attached to keep babies and tots entertained.
- Parents say that the desks allow them to work near their children.
- They're best for short spurts of work, parents say.

When Maegan Moore returned to work about two months ago after the birth of her first child , she found a unique solution for balancing her career with mothering: a work-play desk that lets her care for her baby, Eleanor, while also having a dedicated workspace.

"For her age, it's been awesome," Moore told Business Insider. "I'll try to time it so I can feed her, put her down in the play desk area, and do some work that doesn't involve calls."

The work and play desk is a sort of cubicle designed for parents and children. The desk has a flat work area for parents and a play-pen-like attachment for babies.

Moore uses the desk at a coworking space in New York, but the idea originally started at a library in Virginia that largely serves a disadvantaged population that has trouble accessing both childcare and reliable internet.

Moore doesn't use the desk for long stretches of time — about an hour or two is her current limit. Although she has childcare during the middle of the day, she uses the desk most mornings and afternoons. She says that being able to have Eleanor nearby and nurse her, rather than pumping, has eased her transition back to work.

"That's been a real gift," she said.

## A mom uses the desk to cope with days off school

Bethany Crystal , who contracts for multiple companies in tech and education, uses the work-and-play desk for her 21-month-old, Sydney. Because of the nature of her work, she doesn't have an office to report to. Before finding the work and play desk, she had instances of trying to nurse her baby in a WeWork coworking space or struggling to find a place to put her while she interviewed with firms. The work and play desk has solved that.

"It's really useful to be able to go places for an hour or two and have a place to put a baby," Crystal told BI.

Now that Sydney is a toddler, she's enrolled in a local Montessori school , but the frequent days off mean that Crystal still utilizes the work and play desk regularly.

"There's a lot of inservice days and holidays, which wreaks havoc for professional, entrepreneurial parents like myself," Crystal said.

Sydney is "so happy" to be next to her mother, and Crystal is able to get solid two-hour blocks of work done, which add up over the course of her 60-hour workweek .

Related stories

"Even being able to put her in the crib for little stints made such a big difference," she said.

To Crystal, the desk is representative of a modern work-life balance.

"I believe we are in a new era of work, where it's no longer about what job you want, it's about what kind of lifestyle you want," she said. "For me, it's critically important to have spaces to do work that lean into the messy complications of an imperfect life with a lot of demands."

## Solving the problem of the generation

Both Moore and Crystal use the work and play desk at Workplayce , a family-friendly coworking facility on the Upper West Side of Manhattan. In addition to the desks, the building offers quiet work space, a communal area for parents to work while kids play, and dedicated on-site childcare that can be booked by the hour.

For Crystal, who also has a 4-year-old, the community aspect has been as important as the physical space.

"Not only do I have a place where kids can be kids and I can be at work, but I have fellow parents who are in work-hybrid mode," she said. "It's the first time I've felt like I've been part of a peer-parent community."

"It's refreshing to me," she added, "and helpful for us to see each other and learn how we are all making it work."

When Moore heard about Workplayce, she thought, "This is incredible. What a cool tool to equip working parents."

Prior to this, Moore tried working from home with a babysitter for Eleanor. But living in a small apartment, it was difficult "to not look around and see all the tasks and to-dos," which could distract from work, she said.

Now, she typically goes to the coworking space about four times a week, enrolling Eleanor in childcare when she has a meeting, a call, or work that requires deeper concentration.

"It's great having her around and getting to pop in and see her, but when I need full separate space, I have the ability to do that as well," Moore said.

That setup more accurately matches the lives of many modern working parents, Crystal said.

"There's a unique opportunity for parents to make work and kids fit with their lives. Decoupling work and kids from these arbitrary 9-to-5 work days or 9-to-3 day care days is the first step to making a life that works for you," she said. "This is a really important thing for our generation to solve for."

## Watch: Microsoft's chief brand officer, Kathleen Hall, says the company's employees are its best product testers

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## Solving Law Firm’s Biggest Problem: Outsourcing Legal Billing to Improve Collection and Profitability

- Legal Operations

Streamlining and improving the billing process in law firms is often perceived as a daunting task, fraught with complexities that can challenge even the most seasoned legal professionals. Billing is a time consuming, labor-intensive necessary evil that is often fraught with inaccuracies due to inefficiencies in process and extensive operational silos which can cost a law firm potentially millions of dollars each year.

According to a recent Thomson Reuters survey , the average law firm experiences an 18% realization loss due to billing challenges. This means that nearly one-fifth of billable work is not converted into revenue, negatively impacting the firm’s profitability. 81% of law firms report issues with a significant portion of invoices remaining unpaid or delayed, creating cash flow challenges. This is a huge revenue loss that is creating major problems for law firms, but there are solutions.

## Why is legal billing so complex?

Most unique to attorneys is that their work is often time-consuming difficult to tangibly track. Tracking billable hours accurately requires meticulous attention to detail and can be particularly challenging when dealing with tasks that don't neatly fit into predefined categories. This creates problems because clients often have unique billing preferences and expectations ranging from traditional hourly billing to flat fees, contingency fees and/or blended rates. Accommodating these distinctive preferences for each client and integrating them into a billing system adds a layer of complexity to billing that makes it extremely labor intensive. Additionally, silos that exist within law firms in the form of practice groups or other operational units can make regularity in billing guidelines either difficult to understand or completely nonexistent.

Another complexity in legal billing is the regulatory compliance and ethical guidelines that must be adhered to. Some of these issues include fee agreements, billing transparency, client confidentiality, and conflicts of interest. Staying compliant with these regulations while meeting client expectations and maintaining profitability requires a thorough understanding of legal and professional standards.

While most law firms rely on billing software to streamline the process, integrating these systems with other practice management tools and ensuring data accuracy and security can present technical challenges that require ongoing maintenance and support.

## Reasons to outsource billing

By streamlining the billing processes , law firms can reduce the risk of errors, and focus on core competencies. Technical expertise and best practices ensure process optimization and service quality. Outsourcing can be a valuable solution for law firms aiming to stabilize their cash flow, reduce costs, and improve overall efficiency.

- Access to Advanced Technology: Outsourcing billing often provides access to advanced billing technologies and software platforms that may be cost-prohibitive for smaller law firms to implement independently. These technologies can automate billing processes, provide real-time reporting and analytics, and offer insights into billing trends and patterns that allow firms to optimize their billing practices for greater efficiency and profitability Additionally, predictive AI can provide billing staff with powerful tools to wade through complex billing guidelines in a timely manner, increasing productivity within billing departments.
- Expertise and Accuracy: Billing processes can be complex, especially in law firms where billing requirements may vary from client to client and case to case. Outsourcing billing to specialized professionals ensures that billing is handled by experts who are well-versed in legal billing practices. This expertise leads to greater accuracy in invoicing, reducing the likelihood of errors or discrepancies that could lead to disputes or delays in payment. This also leads to more standardized output.
- Centralization: Often, billing departments in law firms are segregated either geographically or by practice area, leading to inefficiencies and lack of scalability. Centralizing billing can assist by allowing billing to be looked at holistically as a firm rather than piecemeal, leading to a more effective business model for financial processes.
- Efficiency and Timeliness: Outsourcing billing allows law firms to streamline their billing processes and improve efficiency. Billing professionals have the necessary tools and systems in place to generate invoices promptly and ensure timely submission to clients. This helps expedite the payment cycle, leading to improved cash flow for the firm.
- Cost Savings: While some may initially view outsourcing as an additional expense, it often proves to be a cost-effective solution in the long run. By outsourcing billing, law firms eliminate the need to hire and train in-house billing staff, invest in billing software, and bear the overhead costs associated with maintaining billing infrastructure, lowering both soft and hard operational costs and simplifying a billing budget. Additionally, outsourcing billing allows firms to scale their billing operations up or down as needed without incurring significant fixed costs.
- Compliance and Regulation: Legal billing is subject to various regulations and compliance requirements, including those related to client confidentiality, billing transparency, and fee agreements. Outsourcing billing to professionals who are experts in these regulations helps ensure compliance and mitigates the risk of potential legal or ethical issues arising from billing practices.
- Focus on Core Competencies: Attorneys and legal staff excel in providing legal counsel and representation, but are not usually trained in managing billing and administrative tasks, and even if they are, legal staff turning their attention to these tasks from an administrative perspective harms the firm’s bottom line by taking time away from client-facing, billable projects. By outsourcing billing, law firms can free up valuable time and resources to concentrate on what they do best, which is serving clients legal needs.
- Enhanced Client Satisfaction: Timely and accurate billing contributes to a positive client experience. By outsourcing billing , law firms can ensure that clients receive clear, transparent invoices in a timely manner, enhancing client satisfaction and fostering long-term client relationships.

Legal billing is a challenge that costs firms millions of dollars every year. It has become such an issue that many of law firms have said this is their main priority to fix by 2025. Centralized teams face challenges due to the varied requirements of different attorneys and clients. The benefits to outsourcing billing not only will help realization rates but will foster a better relationship with clients and will help alleviate headaches of both lawyers and legal administrators as well.

The contents of this article are intended to convey general information only and not to provide legal advice or opinions.

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Unknown angle problems (with algebra) Google Classroom. Solve for x in the diagram below. x ∘ ( 3 x + 10) ∘. x = ∘. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for ...

Easy mistakes to make. Mixing up the sum of interior angles and exterior angles Students may use 360^{\circ} instead of 180^{\circ} for the sum of the interior angles of the triangle and vice versa.; Equal angles in an isosceles triangle Selecting the wrong angles when identifying the equal angles in an isosceles triangle (particularly a problem when the equal angles are not at the bottom).

If the sum of the measures of two angles is 90∘ 90 ∘, then the angles are complementary. If angle A A and angle B B are complementary, then m∠A+m∠B =90∘ m ∠ A + m ∠ B = 90 ∘. In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications.

angle, right, straight line, point, full turn, vertically, opposite, basic, facts, triangle, quadrilateral

Elena and Diego each wrote equations to represent these diagrams. For each diagram, decide which equation you agree with, and solve it. You can assume that angles that look like right angles are indeed right angles. 1. Elena: \(x=35\) Diego: \(x+35=180\) Figure \(\PageIndex{2}\) 2. Elena: \(35+w+41=180\) Diego: \(w+35=180\) Figure \(\PageIndex ...

The sum of angles on a straight line is 180˚. Interior angles are supplementary. Supplementary angles are angles that add up to 180˚. The sum of angles in a triangle is 180˚. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. The sum of interior angles of a quadrilateral is 360˚.

Linear pair of angles: Sum of angles = 180°. Sum of angles at a point = 360°. 3. Find the measure of an angle which is complementary to 33°. Solution: If the sum of two angles is 90°, they are called complementary angles. Let x be the angle which is complementary to 33°. So, x + 33° = 90°. x = 90° - 33° = 57°.

This lesson first explains supplementary and vertical angles. Then we look at a variety of "angle puzzles", or problems with an unknown angle, and solve those using these basic principles. I also write an equation for each problem, to help students learn to both write and solve equations, since this lesson belongs to my pre-algebra course.

Hand Swap. My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places.

Here we have no equation to solve as the missing angle is 20^{o}. Example 2: angles at a point. Calculate the value of x. Add all known angles. 90+90+120=300. Subtract the angle sum from \bf{360^{o}} . ... Solve the problem using the information you have already gathered.

Detailed solutions and full explanations to grade 8 math questions on angles are presented. Find the unknown angles in the figures below. . Solution. The sum of all 3 interior angles of a triangle is equal to 180°. Hence. 92 + 27 + x = 180. Solve for x. x = 180 - (92 + 27) = 61°.

This means we are given two angles of a triangle and one side, which is the side adjacent to the two given angles. In this case we find the third angle by using Angles of a Triangle, then use The Law of Sines to find each of the other two sides. See Solving "ASA" Triangles. 4. SAS. This means we are given two sides and the included angle.

A straight angle is an angle formed by a pair of opposite rays, or a line. A straight angle has a measure of . A right angle is an angle that is supplementary to itself. A right angle has a measure of . An acute angle has a measure greater than zero but less than that of a right angle, i.e. is acute if and only if .

Find the missing angles in a triangle, around a point, in a quadrilateral, find opposite and supplementary angles, or find angles which require multi-step problem solving skills. For more shape and space resources click here. Scan to open this game on a mobile device. Right-click to copy and paste it onto a homework sheet. Play game ...

Students also learn the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Students are then asked to solve problems related to the exterior angle theorem using Algebra. Example: The measure of the angles of a triangle are in the ratio 2:5:8.

5 Questions. Q1. The exterior angles of a hexagon sum to 540 degrees. Q2. A triangle ALWAYS has each exterior angle as 60 degrees. Q3. The general formula for working out the mean exterior angle of an n-sided polygon is... Q4. The calculation to work out the sum of the interior angles for an octagon would be...

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10•3. Lesson 10: Angle Problems and Solving Equations. Student Outcomes. Students use vertical and adjacent angles and angles on a line and angles at a point in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Start by asking your students if they see angles. Then challenge students to see how many angles they can find. As students find angles, take a marker and highlight at least one angle of each type (acute, right, obtuse, and straight). Draw the marker lines over the angles in the image and identify each one, such as shown below.

Section 7 unravels the intricate tapestry of real-world problem-solving, demonstrating the practical implications of geometric concepts in various fields. Architects employ geometry to design structurally sound buildings, ensuring that shapes, angles, and dimensions harmonize for both aesthetic appeal and stability.

We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $\pi$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid.

Virtual interpolation technology can be applied to direction-of-arrival (DOA) estimation as a preprocessing technique to achieve the DOA estimation for any array. In order to solve the angle-sensitive problem in virtual interpolation technology, a range interpolation transformation method is proposed. Taking the range of the angle function as the interpolating object, and modifying the ...

Solving the problem of the generation. Both Moore and Crystal use the work and play desk at Workplayce, a family-friendly coworking facility on the Upper West Side of Manhattan. In addition to the ...

Angle. Solving Law Firm's Biggest Problem: Outsourcing Legal Billing to Improve Collection and Profitability. Legal Operations; 2 Mins; Streamlining and improving the billing process in law firms is often perceived as a daunting task, fraught with complexities that can challenge even the most seasoned legal professionals. Billing is a time ...