4 2 problem solving angle relationships in triangles answers

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Angle Relationships Simply Explained w/ 11+ Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s lesson, you’re going to learn all about angle relationships and their measures.

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’ll walk through 11 step-by-step examples to ensure mastery.

Let’s dive in!

Angle Pair Relationship Names

In Geometry , there are five fundamental angle pair relationships:

• Complementary Angles
• Supplementary Angles
• Adjacent Angles
• Linear Pair
• Vertical Angles

1. Complementary Angles

Complementary angles are two positive angles whose sum is 90 degrees.

For example, complementary angles can be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two acute angles, like ∠MNP and ∠EFG, whose sum is equal to 90 degrees. Both of these graphics represent pairs of complementary angles.

Complementary Angles Example

2. Supplementary Angles

Supplementary angles are two positive angles whose sum is 180 degrees.

For example, supplementary angles may be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two angles, like ∠MNP and ∠KLR, whose sum is equal to 180 degrees. Both of these graphics represent pairs of supplementary angles.

Supplementary Angles Example

What is important to note is that both complementary and supplementary angles don’t always have to be adjacent angles.

3. Adjacent Angles

Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.

Angles 1 and 2 are adjacent angles because they share a common side.

Adjacent Angles Examples

And as Math is Fun so nicely points out, a straightforward way to remember Complementary and Supplementary measures is to think:

C is for Corner of a Right Angle (90 degrees) S is for Straight Angle (180 degrees)

Now it’s time to talk about my two favorite angle-pair relationships: Linear Pair and Vertical Angles.

4. Linear Pair

A linear pair is precisely what its name indicates. It is a pair of angles sitting on a line! In fact, a linear pair forms supplementary angles.

Because, we know that the measure of a straight angle is 180 degrees, so a linear pair of angles must also add up to 180 degrees.

∠ABD and ∠CBD form a linear pair and are also supplementary angles, where ∠1 + ∠2 = 180 degrees.

Linear Pair Example

5. Vertical Angles

Vertical angles are two nonadjacent angles formed by two intersecting lines or opposite rays.

Think of the letter X. These two intersecting lines form two sets of vertical angles (opposite angles). And more importantly, these vertical angles are congruent.

In the accompanying graphic, we see two intersecting lines, where ∠1 and ∠3 are vertical angles and are congruent. And ∠2 and ∠4 are vertical angles and are also congruent.

Vertical Angles Examples

Together we are going to use our knowledge of Angle Addition, Adjacent Angles, Complementary and Supplementary Angles, as well as Linear Pair and Vertical Angles to find the values of unknown measures.

Angle Relationships – Lesson & Examples (Video)

• Introduction to Angle Pair Relationships
• 00:00:15 – Overview of Complementary, Supplementary, Adjacent, and Vertical Angles and Linear Pair
• Exclusive Content for Member’s Only
• 00:06:29 – Use the diagram to solve for the unknown angle measures (Examples #1-8)
• 00:19:05 – Find the measure of each variable involving Linear Pair and Vertical Angles (Examples #9-12)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Unit 6: Angle relationships

Complementary, supplementary, & vertical angles.

• Angles: introduction (Opens a modal)
• Complementary & supplementary angles (Opens a modal)
• Vertical angles (Opens a modal)
• Complementary and supplementary angles review (Opens a modal)
• Identifying supplementary, complementary, and vertical angles Get 5 of 7 questions to level up!
• Complementary and supplementary angles (visual) Get 3 of 4 questions to level up!
• Complementary and supplementary angles (no visual) Get 5 of 7 questions to level up!
• Vertical angles Get 3 of 4 questions to level up!
• Finding angle measures between intersecting lines Get 3 of 4 questions to level up!

Finding missing angles

• Find measure of vertical angles (Opens a modal)
• Find measure of angles word problem (Opens a modal)
• Equation practice with complementary angles (Opens a modal)
• Equation practice with supplementary angles (Opens a modal)
• Equation practice with vertical angles (Opens a modal)
• Finding missing angles Get 5 of 7 questions to level up!
• Create equations to solve for missing angles Get 5 of 7 questions to level up!
• Unknown angle problems (with algebra) Get 5 of 7 questions to level up!
• Equation practice with angle addition Get 3 of 4 questions to level up!

Triangle angles

• Angles in a triangle sum to 180° proof (Opens a modal)
• Isosceles & equilateral triangles problems (Opens a modal)
• Triangle exterior angle example (Opens a modal)
• Triangle angles review (Opens a modal)
• Find angles in triangles Get 5 of 7 questions to level up!
• Find angles in isosceles triangles Get 3 of 4 questions to level up!
• Triangle exterior angle property problems Get 3 of 4 questions to level up!

Polygon angles

• Sum of interior angles of a polygon (Opens a modal)
• Angles of a polygon Get 3 of 4 questions to level up!

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2.2.4: Solve Right Triangles

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Using the inverse trigonometric functions to solve for missing information about right triangles.

Inverse Trigonometric Ratios

In mathematics, the word inverse means “undo.” For example, addition and subtraction are inverses of each other because one undoes the other. When we use the inverse trigonometric ratios, we can find acute angle measures as long as we are given two sides.

Inverse Tangent : Labeled \(\tan ^{-1}\), the “-1” means inverse.

\(\tan ^{-1} \left(\dfrac{b}{a}\right)=m\angle B\) and \(\tan ^{-1} \left(\dfrac{a}{b}\right)=m\angle A.\)

Inverse Sine : Labeled \(\sin ^{-1}\).

\(\sin ^{-1} \left(\dfrac{b}{c}\right)=m\angle B\) and \(\sin ^{-1} \left(\dfrac{a}{c}\right)=m\angle A.\)

Inverse Cosine : Labeled \(\cos ^{-1}\).

\(\cos ^{-1} \left(\dfrac{a}{c}\right)=m\angle B\) and \(\cos ^{-1} \left(\dfrac{b}{c}\right)=m\angle A.\)

In most problems, to find the measure of the angles you will need to use your calculator. On most scientific and graphing calculators, the buttons look like \([\sin ^{-1}]\), \([\cos ^{-1}]\), and \([\(\tan ^{-1}]\). You might also have to hit a shift or 2nd button to access these functions.

Now that you know both the trig ratios and the inverse trig ratios you can solve a right triangle. To solve a right triangle, you need to find all sides and angles in it. You will usually use sine, cosine, or tangent; inverse sine, inverse cosine, or inverse tangent; or the Pythagorean Theorem .

What if you were told the tangent of \(\angle Z\) is 0.6494? How could you find the measure of \(\angle Z\)?

Example \(\PageIndex{1}\)

Solve the right triangle.

The two acute angles are congruent, making them both \(45^{\circ}\). This is a 45-45-90 triangle. You can use the trigonometric ratios or the special right triangle ratios.

Trigonometric Ratios

\(\begin{array}{rlrl} \tan 45^{\circ} & =\dfrac{15}{B C} & \sin 45^{\circ} & =\dfrac{15}{A C} \\ B C & =\dfrac{15}{\tan 45^{\circ}}=15 & A C & =\dfrac{15}{\sin 45^{\circ}} \approx 21.21 \end{array}\)

45-45-90 Triangle Ratios

\(BC=AB=15 \text{, } AC=15\sqrt{2} \approx 21.21\)

Example \(\PageIndex{2}\)

Use the sides of the triangle and your calculator to find the value of \(\angle A\). Round your answer to the nearest tenth of a degree.

In reference to \(\angle A\), we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.

\(\tan A=\dfrac{20}{25}=\dfrac{4}{5}\). So, \(\tan ^{-1} \dfrac{4}{5}=m\angle A\). Now, use your calculator.

If you are using a TI-83 or 84, the keystrokes would be: [2nd][ TAN ](\(\dfrac{4}{5}\)) [ENTER] and the screen looks like:

\(m\angle A \approx 38.7^{\circ}\)

Example \(\PageIndex{3}\)

\(\angle A\) is an acute angle in a right triangle. Find \(m\angle A\) to the nearest tenth of a degree for \(\sin A=0.68\), \(\cos A=0.85\), and \(\tan A=0.34\).

\(\begin{aligned} m\angle A&=\sin ^{-1} 0.68\approx 42.8^{\circ} \\ m\angle A&=\cos ^{-1} 0.85\approx 31.8^{\circ} \\ m\angle A&=\tan ^{-1} 0.34\approx 18.8^{\circ} \end{aligned}\)

Example \(\PageIndex{4}\)

To solve this right triangle, we need to find \(AB\), \(m\angle C\) and \(m\angle B\). Use only the values you are given.

\(\underline{AB}: \text{ Use the Pythagorean Theorem.}\)

\(\begin{aligned} 24^2+AB^2&=30^2 \\ 576+AB^2&=900 \\ AB^2&=324 \\ AB&=\sqrt{324}=18 \end{aligned}\)

\(\underline{m\angle B} : \text{ Use the inverse sine ratio.}\)

\(\begin{aligned} \sin B &=\dfrac{24}{30}=\dfrac{4}{5} \\ \sin ^{-1} (45) &\approx 53.1^{\circ} =m\angle B\end{aligned}\)

\(\underline{m\angle C} : \text{ Use the inverse cosine ratio.}\)

\(\cos C=\dfrac{24}{30}=\dfrac{4}{5} \rightarrow \cos ^{-1} (\dfrac{4}{5})\approx 36.9^{\circ} =m\angle C\)

Example \(\PageIndex{5}\)

When would you use sin and when would you use \(\sin ^{-1}\) ?

You would use sin when you are given an angle and you are solving for a missing side. You would use \(\sin ^{-1} \)when you are given sides and you are solving for a missing angle.

Use your calculator to find \(m\angle A\) to the nearest tenth of a degree.

Let \(\angle A\) be an acute angle in a right triangle. Find \(m\angle A\) to the nearest tenth of a degree.

• \(\sin A=0.5684\)
• \(\cos A=0.1234\)
• \(\tan A=2.78\)
• \(\cos ^{-1} 0.9845\)
• \(\tan ^{-1} 15.93\)
• \(\sin ^{-1} 0.7851\)

Solving the following right triangles. Find all missing sides and angles. Round any decimal answers to the nearest tenth.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.10.

Additional Resources

Interactive element.

Video: Introduction to Inverse Trigonometric Functions

Activities: Inverse Trigonometric Ratios Discussion Questions

Study Aids: Trigonometric Ratios Study Guide

Practice: Solve Right Triangles

Download 4-2 Reteach Angle Relationships in Triangles

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Into Math Grade 8 Module 4 Lesson 1 Answer Key Develop Angle Relationships for Triangles

We included H MH Into Math Grade 8 Answer Key PDF Module 4 Lesson 1 Develop Angle Relationships for Triangles to make students experts in learning maths.

HMH Into Math Grade 8 Module 4 Lesson 1 Answer Key Develop Angle Relationships for Triangles

I Can find an unknown angle measure in a triangle.

Spark Your Learning

Turn and Talk What conjecture can you make about the sum of the measures of the angles of a triangle?

Build Understanding

B. What do you notice about the sum of the measures of the three triangles? ____________________ Answer: The sum of the measures of the three triangles is 180°

C. Do you think this is true for all triangles? Explain. ____________________

The Triangle Sum Theorem states that the measures of the three interior angles of a triangle sum to 180°.

D. The angles in a triangle measure 2x, 3x, and 4x degrees. Write and solve an equation to determine the angle measures. ____________________ ____________________ Answer: The angles in a triangle measure 2x, 3x, and 4x degrees. The sum of the measures of the three triangles is 180° 2x + 3x + 4x = 180° 9x = 180° x = 180/9 x = 20° 2x = 2 × 20 = 40° 3x = 3 × 20 = 60° 4x = 4 × 20 = 80°

Turn and Talk Discuss how to find a missing measure of an angle in a triangle when the other two angle measures are given.

Step It Out

The Triangle Sum Theorem can be used to draw conclusions about a triangle’s interior angles.

A. What is the sum of the measures of ∠3 and ∠4? __________________ Answer: the sum of the measures of ∠3 and ∠4 is 180°

B. An exterior angle of a polygon is an angle formed by one side of the polygon and the extension of an adjacent side. Which angle in the diagram is an exterior angle? _____________ Answer: ∠4 is an exterior angle.

C. If the measure of ∠3 is 60°, what is the measure of ∠4? _______________________ Answer: ∠3 = 60° ∠3 + ∠4 = 180° 60°+ ∠4 = 180° ∠4 = 180° – 60° ∠4 = 120°

D. If the measure of ∠3 is 60°, what is the sum of the measures of ∠1 and ∠2? ______________ Answer: ∠3 = 60° ∠1 + ∠2 + ∠3 = 180° ∠1 + ∠2 + 60° = 180° ∠1 + ∠2 = 180° – 60° = 120° Thus the sum of the measures of ∠1 and ∠2 is 120°

E. Which angle has a measure equal to the sum of the measures of ∠1 and ∠2? ______________________________ Answer: ∠4 = 120° ∠1 + ∠2 = 180° – 60° = 120° So, ∠4 has a measure equal to the sum of the measures of ∠1 and ∠2.

F. A remote interior angle of an exterior angle of a polygon is an angle that is inside the polygon and is not adjacent to the exterior angle. Which two angles in the diagram are remote interior angles in relation to Angle 4? _____________________________ Answer: ∠1 and ∠2 are the remote interior angles in relation to Angle 4.

G. If the sum of the measures of ∠1 and ∠2 is 115°, what is the measure of ∠4? _______________________ Answer: If the sum of the measures of ∠1 and ∠2 is 115° then the measure of ∠4 is 115°.

Turn and Talk A triangle has exterior Angle P with remote interior Angles Q and R. Can you determine which angle has the greatest measure? Why or why not?

A. Write an equation and solve to find the value of x. Show your work. ___x + ___ = x + ___ __x – x = 80 – ___ x = ___ Answer: 2x + 45° = x + 80° 2x – x = 80° – 45° x = 35°

B. What is the measure of the unknown remote interior angle? _____________________ Answer: the measure of the unknown remote interior angle is 35°

C. Use the value of x from Part A to find the measure of the exterior angle. 2x + 45 = 2(___) + 45 = ___ + 45 = ___ Answer: 2x + 45 2(35) + 45 70° + 45° = 115°

Connect to Vocabulary The measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is the Exterior Angle Theorem.

D. What is the measure of the exterior angle? __________________ Answer: the measure of the exterior angle is 115°

Check Understanding

Question 1. Two angles of a triangle have measures of 30° and 45°. What is the measure of the remaining angle? Answer: Given, Two angles of a triangle have measures of 30° and 45°. Sum of three angles of a triangle = 180° 30°+ 45° + x° = 180° 75° + x° = 180 x° = 180° – 75° x° = 105°

Question 2. Dana draws a triangle with one angle that has a measure of 40°. A. What is the measure of the angle’s adjacent exterior angle? ______________ Answer: Dana draws a triangle with one angle that has a measure of 40°. 180°- 40° = 140° Thus the measure of the angle’s adjacent exterior angle is 140°

B. What is the sum of the measures of the remote interior angles for the exterior angle adjacent to the 40° angle? ______________ Answer: 140° + 40° = 180°

Question 3. An exterior angle of a triangle has a measure of 80°, and one of the remote interior angles has a measure of 20°. Write and solve an equation to find the measure of the other remote interior angle. Answer: Given, An exterior angle of a triangle has a measure of 80°, 180° – 80° = 100° and one of the remote interior angles has a measure of 20°. 180° – 20° – 100° = 60°

On Your Own

Question 4. A puppeteer is making a triangular hat for a puppet. If two of the three angles of the hat both measure 30°, what is the measure of the third angle? Answer: x + 30° + 30° = 180 x + 60° = 180° x = 180° – 60° x = 120° The triangle is an isosceles triangle and the measure of the third angle is 120°

Question 5. Construct Arguments Can a triangle have two obtuse angles? Explain your answer. Answer: No, a triangle does not have two obtuse angles Sum of three angles of a triangle = 180° 100 + 100 = 200° (Not possible)

Question 6. STEM In engineering, equilateral triangles can support the most weight and so are commonly found in the design of bridges and buildings. Equilateral triangles are triangles with three congruent sides and three congruent angles. What are the measures of the angles of an equilateral triangle? Answer: x + x + x = 180° 3x° = 180° x = 180/3 x = 60°

Question 7. A triangle has one 30° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle. Find the measures of the triangle’s unknown angles and explain how you found the answer. Answer: Given, A triangle has one 30° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle. x + 2x + 30° = 180° 3x + 30° = 180° 3x = 180° – 30° 3x = 150° x = 150/3 x = 50° 2x = 2 × 50 = 100°

For Problems 8-10, find the measures of the unknown third angles.

Question 11. Open Ended The measure of an exterior angle of a triangle is x°. The measure of the adjacent interior angle is at least twice x. List three possible solutions for x. Answer: The measure of an exterior angle of a triangle is x°. The measure of the adjacent interior angle is at least twice x. x° + θ = 180° θ = 180° – x ≥ 2x° 180° ≥ 3x° 0° < x ≤ 60° Any three numbers in (0, 60).

Question 12. The measure of an exterior angle of a triangle is 40°. What is the sum of the measures of the corresponding remote interior angles? Answer: The measure of an exterior angle of a triangle is 40°. 2x + 40° = 180° 2x = 180 – 40 2x = 140 x = 140/2 x = 70° Thus the sum of the measures of the corresponding remote interior angles is 140°

I’m in a Learning Mindset!

What did I learn from applying my knowledge of interior angles of a triangle to find the missing exterior angle in Problem 13 that I can explain clearly to a classmate?

Lesson 4.1 More Practice/Homework

Question 3. Construct Arguments Can the measure of an exterior angle of a triangle ever exceed 180? Explain your reasoning. Answer: An exterior angle of a triangle cannot be a straight line because a triangle has 180° in adding all the three angles of a triangle.

Question 5. Open Ended One of the angles in a triangle measures 90°. Name three possibilities for the measures of the remaining two angles. Answer: One of the angles in a triangle measures 90° 30° + 60° + 90° = 180° 90° + 45° + 45° = 180° 90° + 50° + 40° = 180°

Question 8. If an exterior angle of a triangle has a measure of 35°, what is the measure of the adjacent interior angle? Answer: 35° + x = 180° x = 180 – 35 x = 145° Thus the measure of the adjacent interior angle is 145°

Question 10. The measures of an exterior angle of a triangle and its adjacent interior angle add to what value? A. 90° B. 100° C 180° D. 360° Answer: The measures of an exterior angle of a triangle and its adjacent interior angle is equal to 180 degrees. So, option C is the correct answer.

Question 11. The measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles are _____________ Answer: The measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles are 180 degrees.

Spiral Review

Question 12. Hayden and Jamie completed 20 math problems together. Jamie completed 2 more than twice the number that Hayden completed. Let p represent the number of math problems Hayden completed. Write an equation that can be used to find the number of math problems that Jamie completed. Answer: Let p represent the number of Math problems Hayden completed. Let 2p+2 represent the number of Math problems Jamie completed. 2p + 2 + p = 20 3p + 2 = 20 3p = 20 – 2 3p = 18 p = 18/3 = 6 p = 6 Thus Hayden completed 6 math problems. 2p + 2 = 2(6) + 2 = 12 + 2 = 14 Thus Jamie completed 14 Math problems.

Question 13. Does the equation 5(x – 3) = 10x – 15 have one solution, infinitely many solutions, or no solution? Answer: 5(x – 3) = 10x – 15 5x – 15 = 10x – 15 5x – 10x = 15 – 15 -5x = 0 x = 0 Thus x = 0 has infinite number of solutions.

Question 14. Find the value of x, given that 4(3x + 2) = 44. Answer: Given, 4(3x + 2) = 44 12x + 8 = 44 12x = 44 – 8 12x = 36 x = 36/12 x = 3

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7.1.4: Solving for Unknown Angles

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Let's figure out some missing angles.

Exercise \(\PageIndex{1}\): True or False: Length Relationships

Here are some line segments.

Decide if each of these equations is true or false. Be prepared to explain your reasoning.

\(CD+BC=BD\)

\(AB+BD=CD+AD\)

\(AC-AB=AB\)

\(BD-CD=AC-AB\)

Exercise \(\PageIndex{2}\): Info Gap: ANgle Finding

Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.

If your teacher gives you the problem card :

• Silently read your card and think about what information you need to be able to answer the question.
• Ask your partner for the specific information that you need.
• Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem.
• Share the problem card and solve the problem independently.
• Read the data card and discuss your reasoning.

If your teacher gives you the data card :

• Silently read your card.
• Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
• Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
• Read the problem card and solve the problem independently.
• Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Exercise \(\PageIndex{3}\): What's the Match?

Match each figure to an equation that represents what is seen in the figure. For each match, explain how you know they are a match.

• \(g+h=180\)
• \(2h+g=90\)
• \(g+h+48=180\)
• \(g+h+35=180\)

Are you ready for more?

• What is the angle between the hour and minute hands of a clock at 3:00?
• You might think that the angle between the hour and minute hands at 2:20 is 60 degrees, but it is not! The hour hand has moved beyond the 2. Calculate the angle between the clock hands at 2:20.
• Find a time where the hour and minute hand are 40 degrees apart. (Assume that the time has a whole number of minutes.) Is there just one answer?

We can write equations that represent relationships between angles.

• The first pair of angles are supplementary, so \(x+42=180\).
• The second pair of angles are vertical angles, so \(y=28\).
• Assuming the third pair of angles form a right angle, they are complementary, so \(z+64=90\).

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 \(\PageIndex{4}\)

\(M\) is a point on line segment \(KL\). \(NM\) is a line segment. Select all the equations that represent the relationship between the measures of the angles in the figure.

• \(a+b=180\)
• \(180-a=b\)
• \(180=b-a\)

Exercise \(\PageIndex{5}\)

Which equation represents the relationship between the angles in the figure?

• \(88+b=90\)
• \(88+b=180\)
• \(2b+88=90\)
• \(2b+88=180\)

Exercise \(\PageIndex{6}\)

Segments \(AB\), \(EF\), and \(CD\) intersect at point \(C\), and angle \(ACD\) is a right angle. Find the value of \(g\).

Exercise \(\PageIndex{7}\)

Select all the expressions that are the result of decreasing \(x\) by 80%.

• \(\frac{20}{100}x\)
• \(x-\frac{80}{100}x\)
• \(\frac{100-20}{100}x\)
• \((1-0.8)x\)

(From Unit 6.2.6)

Exercise \(\PageIndex{8}\)

Andre is solving the equation \(4(x+\frac{3}{2})=7\). He says, “I can subtract \(\frac{3}{2}\) from each side to get \(4x=\frac{11}{2}\) and then divide by 4 to get \(x=\frac{11}{8}\).” Kiran says, “I think you made a mistake.”

• How can Kiran know for sure that Andre’s solution is incorrect?
• Describe Andre’s error and explain how to correct his work.

(From Unit 6.2.2)

Exercise \(\PageIndex{9}\)

Solve each equation.

\(\begin{array}{lllll}{\frac{1}{7}a+\frac{3}{4}=\frac{9}{8}}&{\qquad}&{\frac{2}{3}+\frac{1}{5}b=\frac{5}{6}}&{\qquad}&{\frac{3}{2}=\frac{4}{3}c+\frac{2}{3}}\\{0.3d+7.9=9.1}&{\qquad}&{11.03=8.78+0.02e}&{\qquad}&{\qquad}\end{array}\)

(From Unit 6.2.1)

Exercise \(\PageIndex{10}\)

A train travels at a constant speed for a long distance. Write the two constants of proportionality for the relationship between distance traveled and elapsed time. Explain what each of them means.

(From Unit 2.2.2)