problem solving for angles

Write the appropriate formula for the situation and substitute in the given information.

[latex]s+40=180[/latex]

[latex]140+40\stackrel{?}{=}180[/latex]

[latex]180=180\checkmark[/latex]

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

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• 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 notation and problem solving

Switch to our new maths teaching resources.

Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom.

Lesson details

Key learning points.

• In this lesson, we will learn about how to problem solve with angles in polygons, along with learning notations for referring to angles.

This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.

Starter quiz

5 questions, lesson appears in, unit maths / angles in polygons.

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.

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Solving Together - Estimating Angles

After you have played some rounds, you might like to discuss why you find some angles easier to estimate than others.  Note for parents: You may find it useful to watch our Guidance for Parents  video for advice on how to get the most out of the Solving Together resources. Estimating Angles enables you to become familiar with angles of different sizes and  builds the important problem solving skills of visualising and estimating skills.

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 )

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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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Course: 7th grade   >   Unit 9

• Find measure of vertical angles

Finding missing angles

• Find measure of angles word problem
• Equation practice with complementary angles
• Equation practice with supplementary angles
• Equation practice with vertical angles
• Create equations to solve for missing angles
• Unknown angle problems (with algebra)

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

• 2 Angle Measure
• 3 Classifying Angles
• 4.1 Properties Used in Angle Chasing

An angle is the union of two rays with a common endpoint . The common endpoint of the rays is called the vertex of the angle, and the rays themselves are called the sides of the angle.

Angle Measure

If two angles are congruent , they have the same angle measure.

A ray drawn from the vertex of the angle, such that the angle formed by this ray and one of the sides is congruent to the angle formed by this ray and the other side, is called the angle bisector .

Classifying Angles

Angle Chasing

Angle chasing is a technique where solvers apply angle properties determine the measures of unknown angles. It is commonly used in geometry problems. Usually this can use a variety of ways including circles , new lines , or transforming the figure somehow. Lots of angle chasing problems require you to think intuitively.

Properties Used in Angle Chasing

• Vertical angles are congruent to each other.
• Parallel lines can create equal or supplementary angles.

• If side lengths are known, the angle bisector theorem can be used to determine that a line bisects an angle.
• If two polygons are congruent , corresponding angles are congruent.
• Finding cyclic quadrilaterals can be a useful strategy in angle chasing since angles opposite with each other are supplementary.

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How To Solve Geometry Problems Involving Angles

Have you ever used a protractor?

If we want to draw an angle, we need a protractor and use it as a guide. Through the protractor, we can visually depict the concept of angles. 

But what exactly is an angle? Why do we have to draw them using protractors? What is their practical significance?

Angles are of particular importance in the study of geometry. They possess potent characteristics that enable us to understand geometric figures better. 

In this module, you will learn everything you must remember about angles, including their definition, classification, and properties.

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Table of Contents

What is an angle.

An angle consists of two rays that have the same endpoint. The endpoint where the rays intersect is called the vertex. Meanwhile, the rays are called the sides .

We use the points of the sides of the angles to name the angle. In the figure above, we call the angle ∠ABC where B is the angle’s vertex (since it is the common endpoint of the rays). We can also name the angle ∠CBA. 

Note that we must put the letter representing the vertex in the middle when naming an angle. 

Be careful when naming an angle. In the figure above, we can call it ∠ABC or ∠CBA but not ∠BAC or ∠CAB since the vertex (point B) must always be in the middle of the angle’s name.  

Sample Problem 1: Determine the angles you can see in the given figure below.

Answer: The angles are ∠PQR, ∠PQS, and ∠SQR.

Sample Problem 2: Using the exact figure above, is SRQ an angle?

Answer : No, because there are no two rays with R as their common endpoint.

Angle Measurement

Just like any geometric figure, we can also measure an angle.

To understand the “measurement of an angle,” we must learn the Protractor Postulate.

Protractor Postulate

“The measurement of an angle refers to the measurement between two rays which  can be designated as a real number from 0 to 180 degrees.”

The protractor postulate assumes that every angle can measure 0 to 180 degrees. Degrees (°) is the unit of measure we use for angles.

How exactly do we measure an angle?

We measure an angle using a protractor. Suppose a protractor was placed against the angle ∠ABC below. We put the ABC vertex in the protractor’s lower middle part. It is clearly seen that ray AB is pointed to 120° while point BC is pointed to 0°. The measurement of ∠ABC is the absolute value of the difference between the numbers the rays are pointed to. Thus, the measure of ∠ABC is 120°

It’s nice to learn how to measure angles using protractors. However, since you are preparing for a college entrance exam (or civil service exam), it is most likely that the measurement of the angles is already given in the questions, so you don’t have to use a protractor to determine the measure of an angle. We just have provided you with an idea of the measurement of the angle. 

We use the symbol m∠ABC to refer to the measurement of angle ∠ABC. Hence, if ∠ABC measures 120°, then m∠ABC = 120°.

Classification of Angles

We can classify angles according to their measurement. An angle can be an acute angle, a right angle, an obtuse angle, or a straight angle.

1. Acute Angle

An acute angle is an angle whose measure is between 0° and 90°.

For instance, if m∠PQR = 45°, ∠PQR is an acute angle.

2. Right Angle

A right angle is an angle whose measure is exactly  90°. Notice how the right angle looks like the letter “L.” 

Take note of the word “exactly” in the definition of right angles. The term “exactly” implies that the measure of a right angle must be strictly 90 degrees. If the angle measure is 90.5°, we cannot consider it a right angle.

3. Obtuse Angle

An obtuse angle is an angle whose measure is between 90° to 180°.

For instance, if m∠XYZ = 105°, ∠XYZ is an obtuse angle.

4. Straight Angle

If an angle has a measurement of exactly 180°, we call that angle a straight angle (which is a straight line).

Sample Problem 1: Determine if the following angles are acute, right, obtuse, or straight.

• m∠ABC = 125°
• m∠COR = 90.5°
• m∠RAW = 0.01°
• m∠CDO = 179.12°
• Obtuse, since 125 is between 90 to 180
• Obtuse, since 90.5 is between 90 to 180 (the angle is not a right angle)
• Acute, since 0.01 is between 0 to 90
• Obtuse, since 179.12 is between 90 to 180

Sample Problem 2: What type of angle is formed by the hands of the clock when it’s three o’clock?

Answer: It is evident that the angle formed by the hands of the clock at three o’clock is a 90° angle since it resembles the letter “L.” Thus, the angle formed is a right angle.

Sample Problem 3: What type of angle is formed by the hands of the clock when it’s two o’clock?

Answer: When the hands of the clock are at three o’clock, they form a right triangle. Since the angle formed when it’s two o’clock is shorter than the angle formed when it’s three o’clock, the angle must be an acute angle.

Angle Addition Postulate

“If D is in the interior of ∠ABC, then the measure of ∠ABC is equal to the sum of the measures of ∠ABD and ∠DBC”

In symbols, m ∠ABC = m∠ABD + m∠DBC

The angle addition postulate is very intuitive and self-explanatory. The concept of the angle addition postulate is analogous to the idea of the segment addition postulate discussed previously. To find the measure of the entire angle ∠ABC in the figure above, we can just add the angles that composed it, which are ∠ ABD and ∠DBC .

Sample Problem 1: If ∠PQR = 25, ∠PQS = 3x + 10, and ∠SQR = 2x, determine the value of x.

The angle addition postulate tells us that the measure of the entire angle ∠PQR is equal to the sum of the measurements of the angles it contains ( ∠ PQS and ∠ SQR).

Hence: m ∠PQR = m∠PQS + m∠SQR

Using the values given in the problem:

m ∠PQR = m∠PQS + m∠SQR

25 = (3x + 10) + 2x

We can now solve the value of x in the given equation above:

25 = 5x + 10 Combining like terms

-10 + 25 = 5x Transposition method

15/5 = 5x/5 Dividing both sides by 5

x = 3 Symmetric property of equality

Thus, the value of x must be 3. 

Sample Problem 2: Using the figure below, determine the measure of ∠ABD.

The only given value is the measure of ∠DBC, which is 52 degrees. 

Notice that the angles ∠ABD and ∠DBC form a straight angle ∠ABC. We know that a straight angle has a measure of 180 degrees. Therefore, using the angle addition postulate:

m ∠ABC = m∠ABD + m∠DBC

Since ∠ABC  is a straight angle:

180 = m∠ABD + m∠DBC

Using the given value of m∠DBC which is 52:

180 =   m∠ABD + 52

Solving for the equation above:

-52 + 180 = m∠ABD 

128 = m∠ABD 

m∠ABD  = 128

Thus, the measure of ∠ABD is 128°.

Congruent Angles

Angles are congruent if they have the exact measurement. This means that congruent angles have the same form and size.  

Do you still remember one of Euclid’s postulates that states that all right angles are congruent ?

It is pretty apparent why the postulate is true. We defined earlier that right angles have a degree measure that is exactly 90°. Thus, all right angles you create will always have a degree measurement of 90°. Hence, concluding that all right angles are congruent is logically sound. 

Angle Bisector

In the figure above, ray QS divides ∠ PQR into two congruent angles: PQS and ∠ SQR. Ray QS is an example of an angle bisector .

An angle bisector is a ray that divides an angle into two congruent angles.

Sample Problem: In the figure below, ∠MNO is 60°. If ray PN bisects angle ∠MNO, what is the measure of ∠ONP?

Since ray PN bisects ∠ MNO, we can conclude that ∠MNO is divided into congruent angles, ∠ MNP and ∠ ONP. This implies that the measurement of angles ∠ MNP and ∠ ONP are equal. 

Since the angles ∠MNP and ∠ ONP have equal measurements, we can determine the measurement of ∠ ONP by dividing the measurement of ∠MNO (60 degrees) by 2.

60 ÷ 2 = 30

Thus, m ∠ ONP is 30 degrees.

Let x represent the measure of ∠ ONP. 

Since ray PN bisects ∠ MNO, then ∠ ONP and ∠ MNP have equal measures.

Thus, the measure of ∠ MNP is also x

By the angle addition postulates:

m ∠ONP + m∠MNP = m∠MNO

x + x = 60 Recall that we let x be the measure of angles ∠ ONP and ∠MNP

2x∕2 = 60/2 Dividing both sides by 2

Since x represents the measure of ∠ ONP, then m ∠ ONP = 30°.

Angle Pairs

As the name suggests, an angle pair consists of two angles that are related in a certain way. In this section, let us discuss each of these angle pairs.

1. Vertical Angles

Vertical angles are formed when two lines intersect. Vertical angles create two pairs of opposite rays.

In the given figure above, ∠ ABC and ∠ DBE are vertical angles. Notice that this angle pair has two pairs of opposite rays: AB and BC (pair 1) and DB and BE (pair 2).

  ∠ ABD and ∠ CBE are also vertical angles (Can you identify the opposite pair of rays?).

However, ∠ ABC and ∠ABD are not vertical angles since they do not form two pairs of opposite rays.

Here’s an “informal” way to detect vertical angles quicker. You can imagine the vertical angles as the opposite openings of the letter “X.” For instance, in the figure above, we can imagine the figure above as the letter X. Notice that angles ∠ABC and ∠DBE are opposite openings of this letter “X,” so they are opposite angles. Same as with ∠ABD and ∠CBE; these angles are also opposite since they are opposite openings of the letter “X.”

Sometimes, vertical angles are also called opposite angles.

Here’s an essential property of vertical angles that you must always keep in mind:

Vertical Angle Theorem

“Vertical angles formed by intersecting lines are always congruent.”

This means that any vertical angle will always have equal degree measurement. 

Retake a look at our previous image:

∠ ABC and  ∠ DBE are vertical angles. As per the vertical angle theorem, we can conclude that they are congruent. For instance, if the measure of ∠ ABC is 45 degrees, m ∠ DBE is also 45 degrees.

Sample Problem: Using the same figure above, if ∠CBE = 2x + 20 and ∠ABD = 120°. What is the value of x?

Solution: We know that angles ∠ ABD and ∠ CBE are vertical angles. Thus, they are congruent or have equal measures:

m ∠ ABD = m ∠ CBE

120 = 2x + 20

2x + 20 = 120 Symmetric property

2x = -20 + 120                       Transposition method

2x/2 = 100/2 

Thus, the value of x is 50.

2. Complementary Angles

If the measurements of two angles have a sum of 90 degrees, the angles are complementary.

∠ PQR and ∠ XYZ are complementary angles since the sum of their measurements is 90 degrees.

3. Supplementary Angles

Supplementary angles are almost similar to complementary angles, except that the sum of their measures must equal 180 degrees. Thus, we can define supplementary angles as a pair of angles whose sum of measurements is 180 degrees. 

In the image above, angles ∠ CAR and ∠ UAV are supplementary angles since the sum of their measurements is 180 degrees. 

Let us solve some problems involving complementary and supplementary angles.

Sample Problem 1: An angle is a complement of another angle. If the measure of one of these angles is twice the measure of the other angle, what is the measure of the shorter angle?

The problem does not provide us with any measurement of the angles. The only thing that we know is that they complement each other. Thus, we can state that the sum of these two angles is 90°.

First angle + Second angle = 90°.

The problem stated that the measure of one angle is twice the other, meaning that one angle is larger in measure than the other. To make our equation above more detailed:

Smaller angle + Larger angle = 90°.

Let x be the measure of the smaller angle. The measure of the larger angle is twice, or two times, the smaller, so we let 2x be the measure of the larger angle:

x + 2x = 90

We can now solve the value of x above:

x + 2x = 90 

3x = 90 Combining like terms

3x∕3 = 90∕3 Dividing both sides of the equation by 3

Since x represents the measure of the smaller angle, the smaller angle has a degree measure of 30 degrees. 

Sample Problem 2: Angles 1 and 2 are supplementary. Angle 1 measures 60 degrees larger than twice the measure of angle 2. What is the measure of an angle that is complementary to angle 2?

The first thing we have to do is determine the measurements of angles 1 and 2.

It states that angles 1 and 2 are supplementary. Hence, the sum of their measurements must be 180:

Angle 1 + Angle 2 = 180 

The measurement of angle 1 is 60 degrees larger than twice the measurement of angle 2. Thus, angle 1 is larger than angle 2 (keep this in mind!)

Let x be the measure of angle 2.

Again, the measurement of angle 1 is 60 degrees larger than twice the measurement of angle 2. We can express the measurement of the angle as 2x + 60 ( 2x is twice the measure of the other angle while the “plus 60 degrees” is for the “60 degrees larger” part of angle 1’s description).

Again, x is the measurement of angle 2. 2x + 60 is the measurement of angle 1:

Going back to the earlier equation we established:

angle 1 + angle 2 = 180 

(2x + 60) + x = 180

Let’s solve for x in the equation we formed above:

3x + 60 = 180 Combining like terms

3x = -60 + 180           Transposition method

3x∕3 = 120∕3 Dividing both sides by 3

Therefore, angle 2 measures 40°.

However, we are not done yet. The problem is not asking us to find the measurement of angle 2; instead, it’s asking us to find the measurement of its complement. The complement of angle 2 is just the angle such that when it is added to the measure of angle 2, the result will be 90°.

So, to find the measurement of the complement of angle 2, we subtract 40° from 90°:

90° – 40° = 50°

So, the final answer for this problem is 50° .

4. Adjacent Angles and Linear Pairs

The angles are adjacent if two angles have a common vertex and a common side (or ray).

In the figure above, ∠ XYZ and ∠ WYZ share a common vertex, point Y, and a side, ray YZ. Thus, ∠ XYZ and ∠ WYZ are adjacent angles.

Now, if two adjacent angles are supplementary, then these angles are called linear pairs. Linear pairs will form a side which is a straight line. 

In the previous figure, ∠ XYZ and ∠ WYZ form a side that is a straight line (line XW). Thus, ∠ XYZ and ∠ WYZ are linear pairs, and they are supplementary. 

Sample Problem: ∠ABC and ∠CBE are linear pairs. Determine the measure of ∠ABC if m∠CBE = 70 degrees. 

Since ∠ABC and ∠CBE are linear pairs, then they are supplementary. We know that the measurements of supplementary angles have a sum of 180 degrees.

To find the measure of ∠CBE:

m∠ABC = 180 – 70 = 110

Thus, the answer is 110 degrees.

Angles Formed by Transversal Intersecting Parallel Lines

As a review, parallel lines do not meet or intersect.

As you can see, lines l 1 and l 2 are parallel since they do not meet. Even if we extend the length of lines infinitely, certainly, they will never intersect.

We use the symbol || to indicate that two lines are parallel. Hence, we can state “line l 1 and l 2 are parallel” in symbols as l 1 || l 2

Now, if a line intersects two parallel lines (refer to the figure below), that line is called a transversal line .

In the figure above, l 3 is a transversal line since it intersects two parallel lines, l 1 and l 2 .

When a transversal line intersects parallel lines, it is noticeable that various angles are formed. Throughout this review, we will use numbers to identify the angles formed by transversal intersecting parallel lines. As you can see, eight angles were created, called transversal angles . The following section will discuss how these transversal angles are related.

1. Corresponding Angles

Corresponding angles are transversal angles on the same side of the transversal and have sides on the transversal going in the same direction. Their other sides go in parallel directions.

In the figure above, angles ∠2 and ∠6 are on the same side of the transversal, with sides on the transversal going in the same direction and their other sides in parallel directions. Likewise, ∠3 and ∠7 are corresponding angles because they exhibit the same properties.

On the other hand, ∠2 and ∠8 are not corresponding angles because although they are on the same side, their figures are not matching.

Here’s an important theorem about corresponding angles.

Corresponding Angle Theorem

“If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.”

So, in the figure above, corresponding angles ∠2 and ∠6 have the exact measurement. So, if m∠2 = 30°, m∠6 should be 30° also. 

Sample Problem 1: In the figure below l 1 || l 2 , if m∠6 = 70°, determine m∠8.

Solution : In the figure above, angles 6 and 8 are corresponding angles since they are on the same “side” (below the transversal) and have matching figures. Therefore, we can state that angles 6 and 8 are congruent. If m∠6 = 70°, then m∠8 = 70°.

Sample Problem 2: Using the exact figure in the previous example, if m∠8 = 70°, determine m∠3.

Solution : If you retake a look at the figure, you will notice that angles ∠3 and ∠8 are vertical since they create two pairs of opposite rays. We know that vertical angles are congruent, so if m∠8 = 70°, then m∠3 = 70°

Sample Problem 3: Using the exact figure in the previous example, if m∠8 = 70°, determine m∠7.

Solution: Notice that angles 7 and 8 are linear pairs since they share a common side, and their sides form a straight line. From our previous discussion about linear pairs, we have learned that linear pairs are supplementary. So, angles 8 and 7 are supplementary angles:

Supplementary angles have a sum of measures of 180 degrees:

 m∠7 +  m∠8 = 180

We know that m∠8 is 70 degrees:

 m∠7 +  70 = 180

Solving for m m∠7:

m∠7 = 180 – 70 = 110°

2. Alternate Interior Angles

Alternate interior angles are transversal angles in the inner portion of the parallel lines but on the opposite side of the transversal.

In the figure above, angles 4 and 5 are alternate interior angles since they are in the interior portion of the parallel lines and on the opposite sides of the transversal (angle 4 is on the right side, angle 5 is on the left side). 

Angles 3 and 6 are also alternate interior angles.

If you have noticed, alternate interior angles form this weird letter “S” shape. If you look again at angles 4 and 5, they form a letter “S”-like figure. Angles 3 and 6 also form the inverse of this letter “S”-like figure. Look at the figure below to better visualize alternate interior angles. 

Here’s an essential theorem about alternate interior angles:

Alternate Interior Angle Theorem

“If two parallel lines are cut by a transversal, the alternate interior angles formed are congruent.”

The theorem above tells us that alternate interior angles have the exact measurement.

Looking at the figure above, we can state that angles 4 and 5 are congruent since they are alternate interior angles. Moreover, we can also say that angles 3 and 6 are congruent since they are alternate interior angles.

3. Alternate Exterior Angles

Alternate exterior angles are the opposite of the alternate interior angles. Alternate exterior angles are a pair of angles in the outer portion of the parallel lines and on the opposite sides of the transversal line.

In the figure above, ∠1 and ∠8 are alternate exterior angles since they are both on the exterior of the parallel lines and the opposite sides of the transversal (∠1 is on the right side of the transversal, and ∠8 is on the left side). 

There’s an important theorem regarding alternate exterior angles. It is stated below.

Alternate Exterior Angle Theorem

“If two parallel lines are cut by a transversal, the alternate exterior angles formed are congruent.”

The theorem above tells us that if two angles are alternate exterior angles, then these angles have an equal measurement or are congruent.

Hence, in the figure above, we can conclude using the theorem that angles 1 and 8 are congruent or have the exact measurement since they are alternate exterior angles.

Sample Problem: Using the given figure below, determine the measurements of angles ∠2, ∠3, ∠6, and ∠8 if m∠1 = 80°

Let us start determining the degree measure of ∠2. Looking at the figure above, you will notice that angles ∠1 and ∠2 are linear pairs since they share a common side, and their remaining sides form a straight line. We know that linear pairs are supplements of each other, so the sum of measures of ∠1 and ∠2 should be 180°:

m∠1 + m∠2 = 180°

It is given that m∠1 = 80°, so let’s plug it into the equation above:

80 + m∠2 = 180° 

m∠2 = 180 – 80 

m∠2 = 100° 

As we can see, the computed measure of ∠2 is 100° or m∠2 = 100°.  

Now, let us determine the measure of ∠3. Retake a look at the given figure. What can you say about ∠1 and ∠3? Yes, they are corresponding angles.

As per the previous theorem we have discussed, corresponding angles are congruent. Since ∠1 and ∠3 are congruent, these angles have the exact measurement. So, if m∠1 = 80° , m∠3 should also equal 80°. Thus, m∠3 = 80 ° .

This time, let us determine the measure of ∠6. Which angle do you think we can use to determine the measurement of ∠6?? Well, you can use either angle 1 or angle 2.

If you use ∠1, then ∠1 and ∠6 are vertical angles. Since vertical angles are congruent (per the vertical angle theorem), if m∠1 = 80°, then m∠6 = 80 ° .

On the other hand, if you use ∠2 instead, ∠2 and ∠6 are linear pairs. Since linear pairs are supplements of each other, then the sum of their measurement is 180° . We have computed earlier that m∠2 = 100°, so to find the measure of ∠6:

m∠6 = 180 –  m∠2 

m∠6 = 180 – 100

Hence, m∠6 = 80° 

Note that whether you use angle 1 or 2, you can still derive the exact measurement for angle 6. 

Lastly, to find the measurement of ∠8, we can use the measurement of ∠1. ∠1 and ∠8 are alternate exterior angles since they are both located in the exterior portion of the parallel lines and are on the opposite sides of the transversal line (look at the given figure above). We know that alternate exterior angles are congruent based on a previous theorem. Hence, if m∠1 = 80°, then m∠8 = 80° .

Interior Angles of a Polygon

Angles that are located inside a polygon are called interior angles. As you may recall, a polygon is a plane figure composed of sides and vertices where these sides meet . Take a look at the triangle ABC (or △ABC) below: 

The triangle above has three interior angles: ∠ABC, ∠ACB, and ∠BAC. We put arcs in the triangle to indicate these interior angles. 

Did you know that if you draw any triangle, the total measurement of its interior angles will always be 180 °?

Yes, the sum of the interior angles of any triangle indeed is 180°. We state this concept formally in the theorem below:

Triangle Sum Theorem

“The sum of the measurements of all the interior angles of any triangle is 180 °.”

So, whether you, your friend, or a stranger draws a triangle, the sum of the interior angles of that triangle will always be precisely 180°. 

Sample Problem: If m∠ABC = 3x + 15, m∠ACB = x + 20, and m∠BAC = x, determine the value of x (refer to the figure below).

Since the given angles are interior angles of the triangle above, we are sure that the sum of the measurements of these angles is 180° because of the triangle sum theorem.

m∠ABC + m∠ACB + m∠BAC = 180°

(3x + 15) + (x + 20) + x = 180° Input the given values in the problem

5x + 35 = 180° Combining like terms

5x = -35 + 180 Transposition method

5x∕5 = 145∕5 Dividing both sides of the equation by 5

Thus, the value of x is 29.

General Formula for the Sum of Interior Angles of a Polygon

A polygon with n sides has n interior angles. So, if a triangle has three sides, then it has three interior angles also. Meanwhile, a square has four sides, so it has four interior angles as well. A pentagon has five sides, so it has five interior angles also.

We have learned in the previous section that the sum of the interior angles of a triangle is always 180°. How about quadrilaterals, pentagons, hexagons, or decagons? How do we find the sum of their interior angles?

We can use a general formula to determine the sum of the interior angles of a polygon with n sides. This formula is presented below:

The formula gives the sum of the interior angles of a polygon with n sides:

Sum of interior angles = 180(n – 2)

So, a quadrilateral that has n = 4 sides has a sum of interior angles:

Sum of interior angles = 180(4 – 2) = 180(2) = 360°

Any quadrilateral (four-sided polygon) such as square, rectangle, parallelogram, trapezoid, etc., will always have a sum of measurements of interior angles equal to 360°

Sample Problem: A polygon has 12 sides (i.e., dodecagon). What is the sum of its interior angles?

Using our formula and n = 12:

Sum of interior angles = 180(12 – 2) 

Sum of interior angles = 180(10) = 1800°

Hence, the sum of the measurements of the interior angles of a dodecagon is 1800°.

Measure of an Interior Angle of a Regular Polygon

As a consequence of the formula above, if a polygon is a regular polygon (which means that all of its sides and angles are congruent), then the measurement of an interior angle of a regular polygon with n sides can be computed as:

Suppose an equilateral and equiangular triangle where all of its sides and angles are congruent. Now, the measure of one of its angles can be calculated using the formula above. 

Using n = 3:

Thus, the measure of an interior angle of an equilateral and equiangular triangle is 60°.

Sample Problem: A regular polygon has n = 10 sides (decagon). Determine the measurement of one of its interior angles.

Using the formula for the measurement of an interior angle of a regular polygon:

Using n = 10:

Thus, the measure of an interior angle of a regular polygon with ten sides (decagon) is 144°.

Perpendicularity

If two lines intersect and form right angles, these lines are perpendicular . In other words, perpendicular lines form right angles.

An informal way to detect perpendicular lines is by looking at the “T shape” formed by these lines. Notice that perpendicular lines form a letter “T” or an inverted letter “T.”

In the figure above, lines l 1 and l 2 are perpendicular. We use the symbol ⊥ to indicate that two lines are perpendicular. Hence, l 1 ⊥ l 2 . Since these lines are perpendicular, ∠ABC and ∠ABE are right angles with m∠ABC and m∠ABE equal 90 degrees.

In addition to this, as you can see above, the angles formed by perpendicular lines are also linear pairs since they share a common side (in the figure above, ray AB) and their remaining sides form a straight line, 

Sample Problem : Lines l 1 and l 2 are perpendicular. If angle 1 measures x degrees and angle 2 measures x + 40 degrees, what is the value of x?

Solution: Since l 1 and l 2 are perpendicular lines, angles 1 and 2 are linear pairs. If two angles are linear pairs, they are supplementary, or the sum of their degree measurements equals 180°.

Thus, we have:

Measurement of angle 1 + Measurement of angle 2 = 180°

(x) + (x + 40) = 180°

Combining like terms:

2x + 40 = 180

By transposition:

2x = -40 + 180

Dividing both sides of the equation by 2:

2x/2 = 140/2

Thus, the value of x is 70. 

Next topic:  Triangles

Previous topic:  Introduction to Geometry: Undefined Terms, Definition, Postulates, and Theorems

Return to the main article:  The Ultimate Basic Math Reviewer

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Affordable Housing Is in Crisis—Can Design Help Solve It?

There is currently an affordable housing shortage in the United States. According to the nonprofit National Low Income Housing Coalition , the nation lacks more than seven million homes for its more than 10.8 million extremely low-income families. And while in the past, this crisis may have only applied to major urban regions, it has expanded such that today, in no state or county can a full-time renter earning minimum wage afford a two-bedroom home.

The problem is quite complex. Historically, the United States Department of Housing and Urban Development provided public housing for those most in need, but in 1998, the Faircloth Amendment capped the number of units that the federal government could construct, resulting in a one-in, one-out policy since. Other subsidies for low-income renters like Section 8 are enacted at the landlord’s discretion, and do not guarantee a unit will be permanently affordable. Thus, the prime way to make up the deficit of units is through conversions or new construction—with developers that can take advantage of tax credits or charity organizations. Add to this equation racist policies like redlining and discriminatory lending, as well as the pandemic-prompted rise in home prices and high mortgage rates that are continuing to ensure only the wealthy have a visible path to homeownership. It’s clear that the systems surrounding affordable housing are broken.

Pico Place, which is located in Santa Monica, California, was designed by Books + Scarpa.

“The lack of affordable housing is a policy problem and a design problem,” argues New York–based architect Alexander Gorlin, co-editor of recent book Housing the Nation: Social Equity, Architecture, and the Future of Affordable Housing . The tome addresses the causes, effects, and potential solutions to this crisis in the United States through a series of essays by designers, economists, community leaders, and others, and then presents a series of successful, attractive, affordable housing projects by architects like Studio Gang , Michael Maltzan Architecture, and more. “Without the policy to build housing there is no design problem,” Gorlin continues, “but design can also come up with solutions to make affordable housing have more potential to build.”

The first step, he says, is bidding on projects that will include public housing units, whether government-sponsored or mixed-income. “I believe there is a moral imperative that more architects and designers should use their talents to help people of all economic levels,” states the architect, who has designed several affordable residential projects across New York with his studio Alexander Gorlin Architects, many providing support housing for formerly unhoused people, foster teens, seniors, or other vulnerable communities.

Rev. Walker's Home designed by Rural Studio.

Though modern architects like Le Corbusier, Bruno Taut, or Walter Gropius might have considered public housing a noble design challenge, many of the best contemporary practitioners prefer to avoid the long, stakeholder-chocked, usually arduous process involved in creating it. However, the times are changing, as visible through high-profile projects like Daniel Libeskind’s newly opened NYCHA Sumner Houses building in Bed-Stuy, Brooklyn , or the buzz around Alabama-based Rural Studio’s innovative Auburn University student design program (also featured in Housing the Nation ), which creates much-needed affordable housing in less populated areas. “It’s not a blight on a community to have one of these buildings,” says Gorlin’s co-editor Victoria Newhouse, an architectural historian. “On the contrary, it can be a great asset.”

Taking on these projects, architects can also identify sticking points in the process and connect with community activists to streamline them. The firm Peterson Rich Office currently consults with the New York City Housing Authority and created a regional plan for design strategies to improve living conditions in public housing in 2020; their process of requiring resident input before construction is one the city continues to this day.

Located in in Washington, D.C., The Aya was designed by Studio Twenty Seven Architecture + Leo A. Daly.

Because of restrictive government policies, getting these well-designed structures built requires convincing developers they are worth the extra steps. At the moment, constructing affordable housing costs just as much as building market rate housing, explains TF Cornerstone senior vice president Jon McMillan in Housing the Nation . Gorlin recommends architects choose economically viable building technologies informed by context—modular, mass timber, 3D-printed, or block and plank construction could be best depending on the region—to help lower the overall costs of the project, providing more incentive for for-profit developers to take it on. Understanding how to spend budget where it has the most impact, like in a cheerful façade or community areas that help create a sense of home, is important, he says. “Being clever, you can achieve the same effect,” as in market rate housing.

Unraveling the web of how and why America’s affordability crisis endures is crucial to understanding how to tackle it, through design and policy. “Our prime goal is to get the book into the hands of people who could make a difference with this situation,” says Newhouse. It’s certainly a topic top of mind for officials. On June 24, Treasury Secretary Janet Yellen announced a three-year, $100 million financial investment to increase affordable housing, following the Biden-Harris Administration’s own proposal last month . The Fed-set interest rates, however, remain unchanged for the moment.

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This Top Warren Buffett Stock Is Helping Microsoft Solve a Big Problem

• Microsoft is signing a landmark carbon removal credit agreement with a subsidiary of Occidental Petroleum.
• The deal will put Microsoft another step closer to achieving its climate goals.
• The agreement is another step toward commercializing Occidental's carbon capture platform.
• Motley Fool Issues Rare “All In” Buy Alert

Occidental Petroleum

Microsoft and Occidental are striving to be part of the solution to climate change.

Carbon dioxide emissions are a big problem. They're a contributing factor to climate change and global warming. It's leading companies worldwide to work hard to reduce their emissions and prevent the rise in global temperatures from exceeding 1.5 degrees Celsius by 2050. That's the level many fear would cause more risks to natural and human systems.

1PointFive Energy, a subsidiary of oil giant Occidental Petroleum ( OXY 0.11% ) , is one of the many companies working toward solutions to prevent the worst impacts of climate change. Occidental, one of Warren Buffett's top holdings , is developing the first of many direct air capture (DAC) systems to pull carbon dioxide from the atmosphere and sequester it underground. The company recently signed a landmark agreement to sell carbon removal credits to Microsoft ( MSFT -2.48% ) , which will help the tech titan reach its climate goals.

Capturing a massive commercial contract

Microsoft wants to be a good corporate citizen. The technology company has committed to becoming carbon-negative by 2030. It has taken several steps to reach that ambitious goal. It recently signed the biggest-ever renewable energy power purchase agreement (PPA) with Brookfield Renewable ( BEPC 2.00% ) ( BEP 3.11% ) . The deal will see Brookfield deliver over 10.5 gigawatts (GW) of new renewable energy capacity to Microsoft in the 2025 to 2030 timeframe. That's almost eight times larger than the biggest corporate PPA ever signed. It also added to the roughly 1 GW Brookfield is already delivering to Microsoft. That deal will help accelerate Brookfield's development pipeline while putting Microsoft closer to achieving 100% of its power needs from zero-carbon energy by the end of the decade, even as its power consumption accelerates due to the massive needs of cloud computing and AI.

However, Microsoft can't achieve its carbon-negative goals on renewable energy alone . That recently led it to sign a landmark agreement with 1PointFive, a carbon capture and storage (CCS) -focused company owned by Occidental Petroleum. It agreed to sell Microsoft 500,000 metric tons of carbon dioxide removal (CDR) credits over six years to help support its carbon removal strategy. That's the biggest-ever purchase of CDR credits enabled by a DAC facility. The landmark agreement will help support Microsoft's commitment to becoming carbon-negative by 2030.

Occidental's subsidiary is building the world's first industrial-scale DAC facility in Texas, called STRATOS. That facility will have the capacity to capture and sequester up to 500,000 tonnes of carbon dioxide per year. Occidental and its partner, Blackrock , expect the project to be commercially operational by the middle of next year. Microsoft's CDR credits will support that facility.

Microsoft joins a growing list of commercial customers committed to supporting STRATOS. 1PointFive has also signed CDR credit purchase agreements with several companies, including Amazon , the Houston Astros, and the Houston Texans.

Capturing a potentially massive opportunity

As a leading oil company, many would likely see Occidental Petroleum as part of the climate change problem. However, the company wants to be part of the solution, seeing this issue as a potentially lucrative opportunity.

The company believes that CCS could grow into a $3 trillion to $5 trillion global market in the coming decades. That's leading it to invest heavily in the technology, which it estimates could eventually contribute as much earnings and cash flow as it currently gets from producing oil and gas.

STRATOS is the first of many DAC facilities Occidental plans to develop. It's already working on an even larger facility. The South Texas Direct Air Capture Hub would feature the first DAC plant capable of removing up to 1 million metric tons of carbon dioxide annually. Meanwhile, that hub could support several facilities that could remove and store up to 30 million metric tons annually.

Occidental also acquired Carbon Engineering for $1.1 billion last year to enhance its ability to develop DAC facilities. That company developed the technology behind its DAC projects. Because Occidetnal now owns Carbon Engineering, it could eventually license this technology to other companies.

CCS represents a massive long-term growth catalyst for Occidental. It enhances the company's already strong near- and medium-term growth drivers, which include its pending acquisition of Crown Rock, increased midstream earnings, and chemicals expansion projects. These growth drivers could add $2 billion to its annual free cash flow total by the second half of 2026. That earnings growth should create a lot of value for investors over the coming years, including leading shareholder Berkshire Hathaway . Buffett's company now owns 28.8% of the oil company's outstanding shares. It's Berkshire's sixth largest holding, at $15.6 billion, accounting for 3.8% of its investment portfolio.

A win-win deal

Microsoft is partnering with a subsidiary of Occidental Petroleum to help it achieve its goal of becoming carbon-negative by 2030. That deal will help the technology company solve a big problem. It's also helping further commercialize Occidental's first large-scale CCS project. That technology could be a needle-moving growth driver, adding to its long-term investment appeal.

John Mackey, former CEO of Whole Foods Market, an Amazon subsidiary, is a member of The Motley Fool's board of directors. Matt DiLallo has positions in Amazon, Berkshire Hathaway, Brookfield Renewable, and Brookfield Renewable Partners. The Motley Fool has positions in and recommends Amazon, Berkshire Hathaway, Brookfield Renewable, and Microsoft. The Motley Fool recommends Brookfield Renewable Partners and Occidental Petroleum and recommends the following options: long January 2026 $395 calls on Microsoft and short January 2026 $405 calls on Microsoft. The Motley Fool has a disclosure policy .

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These are the top skills you need to help land a job in AI

• AI skills are in high demand as companies aim to leverage AI for competitive products.
• AI expertise can lead to lucrative roles in Big Tech or startups and substantial pay raises.
• Here are some of the key skills that employers are seeking for AI-related roles. 

AI skills are in high demand in the job market as more companies seek to use the technology to compete with rivals and become more efficient.

Having AI expertise could also land you a position at a Big Tech giant, startups — or even get you a pay bump .

Nancy Xu, founder and CEO of AI recruitment company Moonhub, told Business Insider that her firm is seeing an uptick in demand for "technical generalists who can build AI applications, along with domain experts in several emerging areas of AI research, large language model training and fine-tuning, and machine learning infrastructure deployment."

She says some companies are going the extra mile to win AI talent. "We're seeing CEOs flying to candidates to close offers, significantly above-average sign-on and performance bonuses, new equity structures, customized benefits for individuals, and more," Xu said.

Iffi Wahla, CEO of global talent network Edge, said data scientists have been among the best-paid tech workers in recent years, partly because every business needs people who can understand and extract value from data. Companies want to spread AI skills across their operations, so those with a background or training in techniques such as prompt engineering on generative AI will benefit from increased job opportunities and pay, Wahla added.

Here are some of the most sought-after skills that may help you land a well-paying job in AI.

Aswini Thota, director of data science at financial service firm USAA, told BI that when hiring data scientists and AI engineers, he assesses candidates based on three key areas: technical prowess, business acumen and communication, and innovation.

Technical knowledge

Thota says data scientists are expected to be well-versed in Python and R, the most popular programming languages for building AI models, while some companies use C++ and Java.

They're also expected to have a foundational knowledge of statistics, as well as machine learning algorithms and frameworks in Python or R.

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"A vast majority of organizations rely on cloud technology to store, analyze, and build models, so a working knowledge of Amazon Web Services, Google Cloud Platform, Snowflake, Microsoft Azure, Databricks, and similar platforms has become increasingly important in recent years," Thota said.

Business acumen and communication

According to Thota, data scientists looking to land a job in AI should also have strong business acumen to grasp an organization's challenges and develop solutions. "Communication skills come into play when data scientists must explain the results and influence decision-makers to align with the technical approach they recommend."

When recruiting for senior or lead roles, Thota says he looks for candidates who have the potential to lead with innovation. "Hiring candidates with an innovative mindset helps us anticipate and address potential challenges before they become issues and also develop groundbreaking solutions."

Flexibility and ongoing learning

Ram Srinivasan, a future of work leader at consulting firm JLL, says some of the most sought-after AI competencies include a combination of technical and soft skills.

They include having a "strong learning mindset and adaptability" because employers look for candidates who can quickly adopt new technologies and methods.

Problem-solving and teamwork

Srinivasan adds: "AI projects often involve complex challenges requiring innovative problem-solving skills. Collaborating effectively with diverse teams, including data scientists, project managers, and product developers, is also essential."

Ethical considerations

AI development poses ethical questions and risks that engineers and developers must navigate responsibly.

Identifying use cases

Srinivasan said tech workers should be able to spot potential AI applications across industries, assess their feasibility, and implement them effectively.

"This involves understanding various sectors, developing implementation strategies, managing organizational change, and measuring ROI. Skills in expanding successful AI pilots and facilitating user adoption are crucial."

Watch: AI expert explains how to incorporate generative AI into your business strategy

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