applying angle relationships homework 2 true or false

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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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Applying Angle Relationships - Solving f...

Mathematics.

Applying Angle Relationships - Solving for X

30 questions

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The two angles shown are:

SUPPLEMENTARY

Which equation best represents the angle relationships shown?

12x + 19 + 22x - 9 = 180

12x + 19 = 22x - 9

What is the value of x?

10x - 20 + 7x + 4 = 180

10x - 20 = 7x + 4

9x + 61 + 3x + 103 = 180

9x + 61 = 3x + 103

(-4x + 5) + (-13x + 39) = 180

-4x + 5 = -13x + 39

7x + 10 + x + 10 = 180

7x + 10 = x + 10

8x + 12x = 180

2x + 54 + 4x - 6 = 180

2x + 54 = 4x - 6

Angle 3 and Angle 6 are examples of which type of angle pair?

Alternate exterior angles

Alternate interior angles

Vertical angles

Corresponding angles

Angle 1 and Angle 5 are examples of which type of angle pair?

Angle 6 and Angle 7 are examples of which type of angle pair?

Which of the following angles would NOT be congruent to the measure of   ∠ 7 \angle7 ∠ 7   ?

  ∠ 2 \angle2 ∠ 2    

  ∠ 3 \angle3 ∠ 3    

  ∠ 6 \angle6 ∠ 6    

  ∠ 8 \angle8 ∠ 8    

Which of the following is an example of corresponding angles?

  ∠ 8   a n d   ∠ 4 \angle8\ and\ \angle4 ∠ 8   a n d   ∠ 4    

  ∠ 5   a n d   ∠ 7 \angle5\ and\ \angle7 ∠ 5   a n d   ∠ 7    

  ∠ 1   a n d   ∠ 7 \angle1\ and\ \angle7 ∠ 1   a n d   ∠ 7    

  ∠ 3   a n d   ∠ 5 \angle3\ and\ \angle5 ∠ 3   a n d   ∠ 5    

Which of the following is NOT an example of supplementary angles?

  ∠ 7   a n d   ∠ 8 \angle7\ and\ \angle8 ∠ 7   a n d   ∠ 8    

  ∠ 2   a n d   ∠ 3 \angle2\ and\ \angle3 ∠ 2   a n d   ∠ 3    

  ∠ 6   a n d   ∠ 4 \angle6\ and\ \angle4 ∠ 6   a n d   ∠ 4    

If the    m ∠ 7   =   115 ° , m\angle7\ =\ 115\degree, m ∠ 7   =   1 1 5 ° ,    find the measure of    ∠ 2. \angle2. ∠ 2 .    

  115 ° 115\degree 1 1 5 °    

  65 ° 65\degree 6 5 °    

  180 ° 180\degree 1 8 0 °    

Cannot be determined

If the    m ∠ 5   =   63 ° , m\angle5\ =\ 63\degree, m ∠ 5   =   6 3 ° ,    find the measure of    ∠ 3. \angle3. ∠ 3 .    

  63 ° 63\degree 6 3 °    

  117 ° 117\degree 1 1 7 °    

If the    m ∠ 1   =   57 ° , m\angle1\ =\ 57\degree, m ∠ 1   =   5 7 ° ,    find the measure of    ∠ 6. \angle6. ∠ 6 .    

  123 ° 123\degree 1 2 3 °    

  57 ° 57\degree 5 7 °    

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Unit 8 Lesson 2 Homework (Applying Angle Relationships)

Angle relationships

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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)