# Big Ideas Math Geometry Answers Chapter 3 Parallel and Perpendicular Lines

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## Big Ideas Math Book Geometry Answer Key Chapter 3 Parallel and Perpendicular Lines

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### Parallel and Perpendicular Lines Maintaining Mathematical Proficiency

Find the slope of the line.

Question 1.

Question 2.

Question 3.

Write an equation of the line that passes through the given point and has the given slope.

Question 4.
(6, 1); m = – 3

Question 5.
(-3, 8); m = – 2

Question 6.
(- 1, 5); m = 4

Question 7.
(2, – 4); m = $$\frac{1}{2}$$

Question 8.
(- 8, – 5); m = –$$\frac{1}{4}$$

Question 9.
(0, 9); m = $$\frac{2}{3}$$

Question 10.
ABSTRACT REASONING
Why does a horizontal line have a slope of 0, but a vertical line has an undefined slope?

### Parallel and Perpendicular Lines Mathematical Practices

Use a graphing calculator to graph the pair of lines. Use a square viewing window. Classify the lines as parallel, perpendicular, coincident, or non perpendicular intersecting lines. Justify your answer.

Question 1.
x + 2y = 2
2x – y = 4

Question 2.
x + 2y = 2
2x + 4y = 4

Question 3.
x + 2y = 2
x + 2y = – 2

Question 4.
x – 2y = 2
x – y = – 4

### 3.1 Pairs of Lines and Angles

Exploration 1

Points of intersection

work with a partner: Write the number of points of intersection of each pair of coplanar lines.

Exploration 2

Classifying Pairs of Lines

Work with a partner: The figure shows a right rectangular prism. All its angles are right angles. Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers. (Two lines are skew lines when they do not intersect and are not coplanar.)

Exploration 3

Identifying Pairs of Angles

Work with a partner: In the figure, two parallel lines are intersected by a third line called a transversal.

a. Identify all the pairs of vertical angles. Explain your reasoning.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.

b. Identify all the linear pairs of angles. Explain your reasoning.

Question 4.
What does it mean when two lines are parallel, intersecting, coincident, or skew?

Question 5.
In Exploration 2. find three more pairs of lines that are different from those given. Classify the pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers.

### Lesson 3.1 Pairs of Lines and Angles

Monitoring Progress

Question 1.
Look at the diagram in Example 1. Name the line(s) through point F that appear skew to .

Question 2.
In Example 2, can you use the Perpendicular Postulate to show that is not perpendicular to ? Explain why or why not.

Classify the pair of numbered angles.

Question 3.

Question 4.

Question 5.

### Exercise 3.1 Pairs of Lines and Angles

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
Two lines that do not intersect and are also not parallel are ________ lines.

Question 2.
WHICH ONE DOESN’T BELONG?
Which angle pair does not belong with the other three? Explain our reasoning.

∠2 and ∠3
∠4 and ∠5
∠1 and ∠8
∠2 and∠7

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, think of each segment in the diagram as part of a line. All the angles are right angles. Which line(s) or plane(s) contain point B and appear to fit the description?

Question 3.
line(s) parallel to .

Question 4.
line(s) PerPendicular to .

Question 5.
line(s) skew to

Question 6.
plane(s) parallel to plane CDH

In Exercises 7-10, Use the diagram.

Question 7.
Name a pair of parallel lines.

Question 8.
Name a pair of perpendicular lines.

Question 9.

Question 10.

In Exercises 11-14, identify all pairs of angles of the given type.

Question 11.
corresponding

Question 12.
alternate interior

Question 13.
alternate exterior

Question 14.
consecutive interior

USING STRUCTURE
In Exercises 15-18, classify the angle pair as corresponding. alternate interior, alternate exterior, or consecutive interior angles.

Question 15.
∠5 and ∠1

Question 16.
∠11 and ∠13

Question 17.
∠6 and ∠13

Question 18.
∠2 and ∠11

ERROR ANALYSIS
In Exercises 19 and 20. describe and correct the error in the conditional statement about lines.

Question 19.

Question 20.

Question 21.
MODELING WITH MATHEMATICS
Use the photo to decide whether the statement is true or false. Explain Your reasoning.

a. The plane containing the floor of the tree house is parallel to the ground.
b. The lines containing the railings of the staircase, such as , are skew to all lines in the plane containing the ground.

c. All the lines containing the balusters. such as , are perpendicular to the plane containing the floor of the tree house.

Question 22.
THOUGHT PROVOKING
If two lines are intersected by a third line, is the third line necessarily a transversal? Justify your answer with a diagram.

Question 23.
MATHEMATICAL CONNECTIONS
Two lines are cut by a transversal. Is it possible for all eight angles formed to have the same measure? Explain your reasoning.

Question 24.
HOW DO YOU SEE IT?
Think of each segment in the figure as part of a line.

a. Which lines are parallel to ?
b. Which lines intersect ?
c. Which lines are skew to ?
d. Should you have named all the lines on the cube in parts (a)-(c) except im – 30? Explain.

In exercises 25-28. copy and complete the statement. List all possible correct answers.

Question 25.
∠BCG and __________ are corresponding angles.

Question 26.
∠BCG and __________ are consecutive interior angles.

Question 27.
∠FCJ and __________ are alternate interior angles.

Question 28.
∠FCA and __________ are alternate exterior angles.

Question 29.
MAKING AN ARGUMENT
Your friend claims the uneven parallel bars in gymnastics are not really Parallel. She says one is higher than the other. so they cannot be in the same plane. Is she correct? Explain.

Maintaining Mathematical Proficiency

Use the diagram to find the measure of all the angles.

Question 30.
m∠1 = 76°

Question 31.
m∠2 = 159°

### 3.2 Parallel Lines and Transversals

Exploration 1

Exploring parallel Lines

Work with a partner: Use dynamic geometry software to draw two parallel lines. Draw a third line that intersects both parallel lines. Find the measures of the eight angles that are formed. What can you conclude?

Exploration 2

Writing conjectures

Work with a partner. Use the results of Exploration 1 to write conjectures about the following pairs of angles formed by two parallel lines and a transversal.
ATTENDING TO PRECISION
To be proficient in math, you need to communicate precisely with others.
a. corresponding angles

b. alternate interior angles

c. alternate exterior angles

d. consecutive interior angles

Question 3.
When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent?

Question 4.
In Exploration 2. m∠1 = 80°. Find the other angle measures.

### Lesson 3.2 Parallel Lines and Transversals

Monitoring Progress

Use the diagram

Question 1.
Given m∠1 = 105°, find m∠4, m∠5, and m∠8. Tell which theorem you use in each case.

Question 2.
Given m∠3 = 68° and m∠8 = (2x + 4)°, what is the value of x? Show your steps.

Question 3.
In the proof in Example 4, if you use the third statement before the second statement. could you still prove the theorem? Explain.

Question 4.
WHAT IF?
In Example 5. yellow light leaves a drop at an angle of m∠2 = 41°. What is m∠1? How do you know?

### Exercise 3.2 Parallel Lines and Transversals

Vocabulary and Core Concept Check

Question 1.
WRITING
How are the Alternate Interior Angles Theorem (Theorem 3.2) and the Alternate Exterior
Angles Theorem (Theorem 3.3) alike? How are they different?

Question 2.
WHICH ONE DOESN’T BELONG?
Which pair of angle measures does not belong with the other three? Explain.

m∠1 and m∠3
m∠2 and m∠4
m∠2 and m∠3
m∠1 and m∠5

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, find m∠1 and m∠2. Tell which theorem you use in each case.

Question 3.

Question 4.

Question 5.

Question 6.

In Exercises 7-10. find the value of x. Show your steps.

Question 7.

Question 8.

Question 9.

Question 10.

In Exercises 11 and 12. find m∠1, m∠2, and m∠3. Explain our reasoning.

Question 11.

Question 12.

Question 13.
ERROR ANALYSIS
Describe and Correct the error in the students reasoning

Question 14.
HOW DO YOU SEE IT?
Use the diagram

a. Name two pairs of congruent angles when $$\ove b. Name two pairs of supplementary angles when [latex]\overline{A B}$$ and $$\overline{D C}$$ are parallel. Explain your reasoning.

PROVING A THEOREM
In Exercises 15 and 16, prove the theorem.

Question 15.
Alternate Exterior Angles Theorem (Thm. 3.3)

Question 16.
Consecutive Interior Angles Theorem (Thm. 3.4)

Question 17.
PROBLEM SOLVING
A group of campers tie up their food between two parallel trees, as shown. The rope is pulled taut. forming a straight line. Find m∠2. Explain our reasoning.

Question 18.
DRAWING CONCLUSIONS
You are designing a box like the one shown.

a. The measure of ∠1 is 70°. Find m∠2 and m∠3.
b. Explain why ∠ABC is a straight angle.
c. If m∠1 is 60°, will ∠ABC still he a straight angle? Will the opening of the box be more steep or less steep? Explain.

Question 19.
CRITICAL THINKING
Is it possible for consecutive interior angles to be congruent? Explain.

Question 20.
THOUGHT PROVOKING
The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible that a transversal intersects two parallel lines? Explain your reasoning.

MATHEMATICAL CONNECTIONS
In Exercises 21 and 22, write and solve a system of linear equations to find the values of x and y.

Question 21.

Question 22.

Question 23.
MAKING AN ARGUMENT
During a game of pool. your friend claims to be able to make the shot Shown in the diagram by hitting the cue ball so that m∠1 = 25°. Is your friend correct? Explain your reasoning.

Question 24.
REASONING
In the diagram. ∠4 ≅∠5 and $$\overline{S E}$$ bisects ∠RSF. Find m∠1. Explain your reasoning.

Maintaining Mathematical Proficiency

Write the converse of the conditional statement. Decide whether it is true or false.

Question 25.
If two angles are vertical angles. then they are congruent.

Question 26.
If you go to the zoo, then you will see a tiger.

Question 27.
If two angles form a linear pair. then they are supplementary.

Question 28.
If it is warm outside, then we will go to the park.

### 3.3 Proofs with Parallel Lines

Exploration 1

Exploring Converses

Work with a partner: Write the converse of each conditional statement. Draw a diagram to represent the converse. Determine whether the converse is true. Justify your conclusion.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

a. Corresponding Angles Theorem (Theorem 3.1): If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Converse:
________________________________

________________________________

________________________________

b. Alternate Interior Angles Theorem (Theorem 3.2): If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Converse:
________________________________

________________________________

________________________________

c. Alternate Exterior Angles Theorem (Theorem 3.3): If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

________________________________

________________________________

________________________________

d. Consecutive Interior Angles Theorem (Theorem 3.4): If two parallel lines are cut by a transversal. then the pairs of consecutive interior angles are supplementary.

________________________________

________________________________

________________________________

Question 2.
For which of the theorems involving parallel lines and transversals is the converse true?

Question 3.
In Exploration 1, explain how you would prove any of the theorems that you found to be true.

### Lesson 3.3 Proofs with Parallel Lines

Monitoring Progress

Question 1.
Is there enough information in the diagram to conclude that m || n? Explain.

Question 2.
Explain why the Corresponding Angles Converse is the converse of the Corresponding Angles Theorem (Theorem 3.1).

Question 3.
If you use the diagram below to prove the Alternate Exterior Angles Converse. what Given and Prove statements would you use?

Question 4.
Copy and complete the following paragraph proof of the Alternate Interior Angles Converse using the diagram in Example 2.

It is given that ∠4 ≅∠5. By the _______ . ∠1 ≅ ∠4. Then by the Transitive Property of Congruence (Theorem 2.2), _______ . So, by the _______ , g || h.

Question 5.
Each step is parallel to the step immediately above it. The bottom step is parallel to the ground. Explain why the top step is parallel t0 the ground.

Question 6.
In the diagram below. p || q and q || r. Find m∠8. Explain your reasoning.

### Exercise 3.3 Proofs with Parallel Lines

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Two lines are cut by a transversal. Which angle pairs must be congruent for the lines to be parallel?

Question 2.
WRITING
Use the theorems from Section 3.2 and the converses of those theorems in this section to write three biconditional statements about parallel lines and transversals.

Monitoring Progress and Modeling with Mathematics

In Exercises 3-8. find the value of x that makes m || n. Explain your reasoning.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

In Exercises 9 and 10, use a compass and straightedge to construct a line through point P that is parallel to line m.

Question 9.

Question 10.

PROVING A THEOREM
In Exercises 11 and 12. prove the theorem.
Question 11.
Alternate Exterior Angles Converse (Theorem 3.7)

Question 12.
Consecutive Interior Angles Converse (Theorem 3.8)

In Exercises 13-18. decide whether there is enough information to prove that m || n. If so, state the theorem you would use.

Question 13.

Question 14.

Question 15.

Question 16.

Question 17.

Question 18.

ERROR ANALYSIS
In Exercises 19 and 20, describe and correct the error in the reasoning.

Question 19.

Question 20.

In Exercises 21-24. are and parallel? Explain your reasoning.

Question 21.

Question 22.

Question 23.

Question 24.

Question 25.
ANALYZING RELATIONSHIPS
The map shows part of Denser, Colorado, Use the markings on the map. Are the numbered streets parallel to one another? Explain your reasoning.

Question 26.
ANALYZING RELATIONSHIPS
Each rung of the ladder is parallel to the rung directly above it. Explain why the top rung is parallel to the bottom rung.

Question 27.
MODELING WITH MATHEMATICS
The diagram of the control bar of the kite shows the angles formed between the Control bar and the kite lines. How do you know that n is parallel to m?

Question 28.
REASONING
Use the diagram. Which rays are parallel? Which rays arc not parallel? Explain your reasoning.

Question 29.
ATTENDING TO PRECISION
Use the diagram. Which theorems allow you to conclude that m || n? Select all that apply. Explain your reasoning.

(A) Corresponding Angles Converse (Thm 3.5)
(B) Alternate Interior Angles Converse (Thm 3.6)
(C) Alternate Exterior Angles Converse (Thm 3.7)
(D) Consecutive Interior Angles Converse (Thm 3.8)

Question 30.
MODELING WITH MATHEMATICS
One way to build stairs is to attach triangular blocks to an angled support, as shown. The sides of the angled support are parallel. If the support makes a 32° angle with the floor, what must m∠1 so the top of the step will be parallel to the floor? Explain your reasoning.

Question 31.
ABSTRACT REASONING
In the diagram, how many angles must be given to determine whether j || k? Give four examples that would allow you to conclude that j || k using the theorems from this lesson.

Question 32.
THOUGHT PROVOKING
Draw a diagram of at least two lines cut by at least one transversal. Mark you diagram so that it cannot be proven that any lines are parallel. Then explain how your diagram would need to change in order to prove that lines are parallel.

PROOF
In Exercises 33-36, write a proof.

Question 33.
Given m∠1 = 115°, m∠2 = 65°
Prove m||n

Question 34.
Given ∠1 and ∠3 are supplementary.
Prove m||n

Question 35.
Given ∠1 ≅ ∠2, ∠3 ≅ ∠4
Prove $$\overline{A B} \| \overline{C D}$$

Question 36.
Given a||b, ∠2 ≅ ∠3
Prove c||d

Question 37.
MAKING AN ARGUMENT

Question 38.
HOW DO YOU SEE IT?
Are the markings on the diagram enough to conclude that any lines are parallel? If so. which ones? If not, what other information is needed?

Question 39.
PROVING A THEOREM
Use these steps to prove the Transitive Property of Parallel Lines Theorem
a. Cops the diagram with the Transitive Property of Parallel Lines Theorem on page 141.
b. Write the Given and Prove statements.
c. Use the properties of angles formed by parallel lines cut by a transversal to prove the theorem.

Question 40.
MATHEMATICAL CONNECTIONS
Use the diagram

a. Find the value of x that makes p || q.
b. Find the value of y that makes r || s.
c. Can r be parallel to s and can p, be parallel to q at the same time? Explain your reasoning.

Maintaining Mathematical Proficiency
Use the Distance Formula to find time distance between the two points.

Question 41.
(1, 3) and (- 2, 9)

Question 42.
(- 3, 7) and (8, – 6)

Question 43.
(5, – 4) and (0, 8)

Question 44.
(13, 1) and (9, – 4)

### 3.1 – 3.3 Study Skills: Analyzing Your Errors

Mathematical Practices

Question 1.
Draw the portion of the diagram that you used to answer Exercise 26 on page 130.

Question 2.
In Exercise 40 on page 144. explain how you started solving the problem and why you started that way.

### 3.1 – 3.3 Quiz

Think of each segment in the diagram as part of a line. Which lines(s) or plane(s) contain point G and appear to fit the description?

Question 1.
line(s) parallel to .

Question 2.
line(s) perpendicular to .

Question 3.
line(s) skew to .

Question 4.

Identify all pairs of angles of the given type.

Question 5.
consecutive interior

Question 6.
alternate interior

Question 7.
corresponding

Question 8.
alternate exterior

Find m∠1 and m∠2. Tell which theorem you use in each case.

Question 9.

Question 10.

Question 11.

Decide whether there is enough information to prove that m || n. If so, state the theorem you would use.

Question 12.

Question 13.

Question 14.

Question 15.
Cellular phones use bars like the ones shown to indicate how much signal strength a phone receives from the nearest service tower. Each bar is parallel to the bar directly next to it.

a. Explain why the tallest bar is parallel to the shortest bar.

b. Imagine that the left side of each bar extends infinitely as a line.
If m∠1 = 58°, then what is m∠2?

Question 16.
The diagram shows lines formed on a tennis court.

a. Identify two pairs of parallel lines so that each pair is in a different plane.

b. Identify two pairs of perpendicular lines.

c. Identify two pairs of skew line

d. Prove that ∠1 ≅ ∠2.

### 3.4 Proofs with Perpendicular Lines

Exploration 1

Writing Conjectures

Work with a partner: Fold a piece of pair in half twice. Label points on the two creases. as shown.

a. Write a conjecture about $$\overline{A B}$$ and $$\overline{C D}$$. Justify your conjecture.

b. Write a conjecture about $$\overline{A O}$$ and $$\overline{O B}$$ Justify your conjecture.

Exploration 2

Exploring a segment Bisector

Work with a partner: Fold and crease a piece of paper. as shown. Label the ends of the crease as A and B.

a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold.

b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles?

Exploration 3

Writing a conjecture

Work with a partner.

a. Draw $$\overline{A B}$$, as shown.

b. Draw an arc with center A on each side of AB. Using the same compass selling, draw an arc with center B on each side $$\overline{A B}$$. Label the intersections of the arcs C and D.

c. Draw $$\overline{C D}$$. Label its intersection with $$\overline{A B}$$ as 0. Write a conjecture about the resulting diagram. Justify your conjecture.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Question 4.
What conjectures can you make about perpendicular lines?

Question 5.
In Exploration 3. find AO and OB when AB = 4 units.

### Lesson 3.4 Proofs with Perpendicular Lines

Monitoring Progress

Question 1.
Find the distance from point E to

Question 2.
Prove the Perpendicular Transversal Theorem using the diagram in Example 2 and the Alternate Exterior Angles Theorem (Theorem 3.3).

Question 3.
Is b || a? Explain your reasoning.

Question 4.
Is b ⊥ c? Explain your reasoning.

### Exercise 3.4 Proofs with Perpendicular Lines

Vocabulary and core Concept Check

Question 1.
COMPLETE THE SENTENCE
The perpendicular bisector of a segment is the line that passes through the _______________ of the segment at a _______________ angle.

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Find the distance from point X to

Find XZ

Find the length of $$\overline{X Y}$$

Find the distance from line l to point X.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4. find the distance from point A to .

Question 3.

Question 4.

CONSTRUCTION
In Exercises 5-8, trace line m and point P. Then use a compass and straightedge to construct a line perpendicular to line m through point P.

Question 5.

Question 6.

Question 7.

Question 8.

CONSTRUCTION
In Exercises 9 and 10, trace $$\overline{A B}$$. Then use a compass and straightedge to construct the perpendicular bisector of $$\overline{A B}$$

Question 9.

Question 10.

ERROR ANALYSIS
In Exercises 11 and 12, describe and correct the error in the statement about the diagram.
Question 11.

Question 12.

PROVING A THEOREM
In Exercises 13 and 14, prove the theorem.
Question 13.
Linear Pair Perpendicular Theorem (Thm. 3. 10)

Question 14.
Lines Perpendicular to a Transversal Theorem (Thm. 3.12)

PROOF
In Exercises 15 and 16, use the diagram to write a proof of the statement.

Question 15.
If two intersecting lines are perpendicular. then they intersect to form four right angles.
Given a ⊥ b
Prove ∠1, ∠2, ∠3, and ∠4 are right angles.

Question 16.
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Given $$\overrightarrow{B A}$$ ⊥$$\vec{B}$$C
Prove ∠1 and ∠2 are complementary

In Exercises 17-22, determine which lines, if any, must be parallel. Explain your reasoning.

Question 17.

Question 18.

Question 19.

Question 20.

Question 21.

Question 22.

Question 23.
USING STRUCTURE
Find all the unknown angle measures in the diagram. Justify your answer for cacti angle measure.

Question 24.
MAKING AN ARGUMENT
Your friend claims that because you can find the distance from a point to aline, you should be able to find the distance between any two lines. Is your friend correct? Explain your reasoning.

Question 25.
MATHEMATICAL CONNECTIONS
Find the value of x when a ⊥ b and b || c.

Question 26.
HOW DO YOU SEE IT?
You are trying to cross a stream from point A. Which point should you jump to in order to jump the shortest distance? Explain your reasoning.

Question 27.
ATTENDING TO PRECISION
In which of the following diagrams is $$\overline{A C}$$ || $$\overline{B D}$$ and $$\overline{A C}$$ ⊥ $$\overline{C D}$$? Select all that apply.
(A)
(B)
(C)
(D)
(E)

Question 28.
THOUGHT PROVOKING
The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry. how many right angles are formed by two perpendicular lines? Justify your answer.

Question 29.
CONSTRUCTION
Construct a square of side length AB

Question 30.
ANALYZING RELATIONSHIPS
The painted line segments that brain the path of a crosswalk are usually perpendicular to the crosswalk. Sketch what the segments in the ph0t0 would look like if they were perpendicular to the crosswalk. Which type of line segment requires less paint? Explain your reasoning.

Question 31.
ABSTRACT REASONING
Two lines, a and b, are perpendicular to line c. Line d is parallel to line c. The distance between lines a and b is x meters. The distance between lines c and d is y meters. What shape is formed by the intersections of the four lines?

Question 32.
MATHEMATICAL CONNECTIONS
Find the distance between the lines with the equations y = $$\frac{3}{2}$$ + 4 and – 3x + 2y = – 1.

Question 33.
WRITING
Describe how you would find the distance from a point to a plane. Can you find the distance from a line to a plane? Explain your reasoning.

Maintaining Mathematical Proficiency

Simplify the ratio.

Question 34.
$$\frac{6-(-4)}{8-3}$$

Question 35.
$$\frac{3-5}{4-1}$$

Question 36.
$$\frac{8-(-3)}{7-(-2)}$$

Question 37.
$$\frac{13-4}{2-(-1)}$$

Identify the slope and they y-intercept of the line.

Question 38.
y = 3x + 9

Question 39.
y = –$$\frac{1}{2}$$x + 7

Question 40.
y = $$\frac{1}{6}$$x – 8

Question 41.
y = – 8x – 6

### 3.5 Equations of Parallel and Perpendicular Lines

Exploration 1

Writing Equations of Parallel and Perpendicular Lines

Work with a partner: Write an equation of the line that is parallel or perpendicular to the given line and passes through the given point. Use a graphing calculator to verify your answer. What is the relationship between the slopes?
a.

b.

c.

d.

e.

f.

Exploration 2

Writing Equations of Parallel and Perpendicular Lines

Work with a partner: Write the equations of the parallel or perpendicular lines. Use a graphing calculator to verify your answers.

a.

b.

Question 3.
How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point?
MODELING WITH MATHEMATICS
To be proficient in math, you need to analyze relationships mathematically to draw conclusions.

Question 4.
Write an equation of the line that is (a) parallel and (b) perpendicular to the line y = 3x + 2 and passes through the point (1, -2).

### Lesson 3.5 Equations of Parallel and Perpendicular Lines

Monitoring Progress

Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio.

Question 1.
A(1, 3), B(8, 4); 4 to 1

Question 2.
A(- 2, 1), B(4, 5); 3 to 7

Question 3.
Determine which of the lines are parallel and which of the lines are perpendicular.

Question 4.
Write an equation of the line that passes through the point (1, 5) and is
(a) parallel to the line y = 3x – 5 and

(b) perpendicular to the line y = 3x – 5.

Question 5.
How do you know that the lines x = 4 and y = 2 are perpendicular?

Question 6.
Find the distance from the point (6, 4) to the line y = x + 4.

Question 7.
Find the distance from the point (- 1, 6) to the line y = – 2x.

### Exercise 3.5 Equations of Parallel and Perpendicular Lines

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
A _________ line segment AB is a segment that represents moving from point A to point B.

Question 2.
WRITING
How are the slopes of perpendicular lines related?

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6. find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio.

Question 3.
A(8, 0), B(3, – 2); 1 to 4

Question 4.
A(- 2, – 4), B(6, 1); 3 to 2

Question 5.
A(1, 6), B(- 2, – 3); 5 to 1

Question 6.
A(- 3, 2), B(5, – 4); 2 to 6

In Exercises 7 and 8, determine which of the lines are parallel and which of the lines are perpendicular.

Question 7.

Question 8.

In Exercises 9 – 12, tell whether the lines through the given points are parallel, perpendicular, or neither. justify your answer.

Question 9.
Line 1: (1, 0), (7, 4)
Line 2: (7, 0), (3, 6)

Question 10.
Line 1: (- 3, 1), (- 7, – 2)
Line 2: (2, – 1), (8, 4)

Question 11.
Line 1: (- 9, 3), (- 5, 7)
Line 2: (- 11, 6), (- 7, 2)

Question 12.
Line 1: (10, 5), (- 8, 9)
Line 2: (2, – 4), (11, – 6)

In Exercises 13 – 16. write an equation of the line passing through point P that ¡s parallel to the given line. Graph the equations of the lines to check that they are parallel.

Question 13.
P(0, – 1), y = – 2 + 3

Question 14.
P(3, 8), y = $$\frac{1}{5}$$(x + 4)

Question 15.
P(- 2, 6), x = – 5

Question 16.
P(4, 0), – x + 2y = 12

In Exercises 17 – 20. write an equation of the line passing through point P that is perpendicular to the given line. Graph the equations of the lines to check that they are perpendicular.

Question 17.
P(0, 0), y = – 9x – 1

Question 18.
P(4, – 6)y = – 3

Question 19.
P(2, 3), y – 4 = – 2(x + 3)

Question 20.
P(- 8, 0), 3x – 5y = 6

In Exercises 21 – 24, find the distance from point A to the given line.

Question 21.
A(- 1, 7), y = 3x

Question 22.
A(- 9, – 3), y = x – 6

Question 23.
A(15, – 21), 5x + 2y = 4

Question 24.
A(- $$\frac{1}{4}$$, 5), – x + 2y = 14

Question 25.
ERROR ANALYSIS
Describe and correct the error in determining whether the lines are parallel. perpendicular, or neither.

Question 26.
ERROR ANALYSIS
Describe and correct the error in writing an equation of the line that passes through the point (3, 4) and is parallel to the line y = 2x + 1.

In Exercises 27-30. find the midpoint of $$\overline{P Q}$$. Then write
an equation of the line that passes through the midpoint and is perpendicular to $$\overline{P Q}$$. This line is called the perpendicular bisector.

Question 27.
P( – 4, 3), Q(4, – 1)

Question 28.
P(- 5, – 5), Q(3, 3)

Question 29.
P(0, 2), Q(6, – 2)

Question 30.
P(- 7, 0), Q(1, 8)

Question 31.
MODELING WITH MATHEMATICS
Your school lies directly between your house and the movie theater. The distance from your house to the school is one-fourth of the distance from the school to the movie theater. What point on the graph represents your school?

Question 32.
REASONING

Question 33.
REASONING
A triangle has vertices L(0, 6), M(5, 8). and N(4, – 1), Is the triangle a right triangle? Explain ‘your reasoning.

Question 34.
MODELING WITH MATHEMATICS
A new road is being constructed parallel to the train tracks through points V. An equation of the line representing the train tracks is y = 2x. Find an equation of the line representing the new road.

Question 35.
MODELING WITH MATHEMATICS
A bike path is being constructed perpendicular to Washington Boulevard through point P(2, 2). An equation of the line representing Washington Boulevard is y = –$$\frac{2}{3}$$x. Find an equation of the line representing the bike path.

Question 36.
PROBLEM SOLVING
A gazebo is being built near a nature trail. An equation of the line representing the nature trail is y = $$\frac{1}{3}$$x – 4. Each unit in the coordinate plane corresponds to 10 feet. Approximately how far is the gazebo from the nature trail?

Question 37.
CRITICAL THINKING
The slope of line l is greater than 0 and less than 1. Write an inequality for the slope of a line perpendicular to l. Explain your reasoning.

Question 38.
HOW DO YOU SEE IT?

Question 39.
CRITICAL THINKING
Suppose point P divides the directed line segment XY So that the ratio 0f XP to PY is 3 to 5. Describe the point that divides the directed line segment YX so that the ratio of YP Lo PX is 5 to 3.

Question 40.
MAKING AN ARGUMENT
Your classmate claims that no two nonvertical parallel lines can have the same y-intercept. Is your classmate correct? Explain.

Question 41.
MATHEMATICAL CONNECTIONS
Solve each system of equations algebraically. Make a conjecture about what the solution(s) can tell you about whether the lines intersect. are parallel, or are the same line.
a. y = 4x + 9
4x – y = 1
b. 3y + 4x = 16
2x – y = 18
c. y = – 5x + 6
10x + 2y = 12

Question 42.
THOUGHT PROVOKING
Find a formula for the distance from the point (x0 V0) to the line ax + by = 0. Verify your formula using a point and a line.

MATHEMATICAL CONNECTIONS
In Exercises 43 and 44, find a value for k based on the given description.

Question 43.
The line through (- 1, k) and (- 7, – 2) is parallel to the line y = x + 1.

Question 44.
The line through (k, 2) and (7, 0) is perpendicular to the line y = x – $$\frac{28}{5}$$.

Question 45.
ABSTRACT REASONING
Make a conjecture about how to find the coordinates of a point that lies beyond point B along $$\vec{A}$$B. Use an example to support your conjecture.

Question 46.
PROBLEM SOLVING
What is the distance between the lines y = 2x and y = 2x + 5? Verify your answer.

PROVING A THEOREM
In Exercises 47 and 48, use the slopes of lines to write a paragraph proof of the theorem.

Question 47.
Lines Perpendicular to a Transversal Theorem (Theorem 3.12): In a plane. if two lines are perpendicular to the same line. then they are parallel to each other.

Question 48.
Transitive Property of Parallel Lines Theorem (Theorem 3.9): If two lines are parallel to the same line, then they are parallel to each other.

Question 49.
PROOF
Prove the statement: If two lines are vertical. then they are parallel.

Question 50.
PROOF
Prove the statement: If two lines are horizontal, then they are parallel.

Question 51.
PROOF
Prove that horizontal lines are perpendicular to vertical lines.

Maintaining Mathematical Proficiency

Plot the point in a coordinate plane.

Question 52.
A(3, 6)

Question 53.
B(0, – 4)

Question 54.
C(5, 0)

Question 55.
D( – 1, – 2)

Copy and complete the table.

Question 56.

Question 57.

### 3.4 – 3.5 Performance Task: Navajo Rugs

Mathematical Practices

Question 1.
Compare the effectiveness of the argument in Exercise 24 on page 153 with the argument “You can find the distance between any two parallel lines” What flaw(s) exist in the argument(s)? Does either argument use correct reasoning? Explain.

Question 2.
Look back at your construction of a square in Exercise 29 on page 154. How would your
construction change if you were to construct a rectangle?

Question 3.
In Exercise 31 on page 161, a classmate tells you that our answer is incorrect because you should have divided the segment into four congruent pieces. Respond to your classmates argument by justifying your original answer.

### Parallel and Perpendicular Lines Chapter Review

#### 3.1 Pairs of Lines and Angles

Think of each segment in the figure as part of a line. Which line(s) or plane(s) appear to fit the description?

Question 1.
line(s) perpendicular to

Question 2.
line(s) parallel to

Question 3.
line(s) skew to

Question 4.
plane(s) parallel to plane LMQ

#### 3.2 Parallel Lines and Transversals

Find the values of x and y.

Question 5.

Question 6.

Question 7.

Question 8.

#### 3.3 Proofs with Parallel Lines

Find the value of x that makes m || n.

Question 9.

Question 10.

Question 11.

Question 12.

#### 3.4 Proofs with Perpendicular Lines

Determine which lines, if any, must be parallel. Explain your reasoning.

Question 13.

Question 14.

Question 15.

Question 16.

#### 3.5 Equations of Parallel and Perpendicular Lines

Write an equation of the line passing through the given point that is parallel to the given line.

Question 17.
A(3, – 4),y = – x + 8

Question 18.
A(- 6, 5), y = $$\frac{1}{2}$$x – 7

Question 19.
A(2, 0), y = 3x – 5

Question 20.
A(3, – 1), y = $$\frac{1}{3}$$x + 10

Write an equation of the line passing through the given point that is perpendicular to the given line.

Question 21.
A(6, – 1), y = – 2x + 8

Question 22.
A(0, 3), y = – $$\frac{1}{2}$$x – 6

Question 23.
A(8, 2),y = 4x – 7

Question 24.
A(-1, 5), y = $$\frac{1}{7}$$x + 4

Find the distance front point A to the given line.

Question 25.
A(2, – 1), y = – x + 4

Question 26.
A(- 2, 3), y = $$\frac{1}{2}$$x + 1

### Parallel and Perpendicular Lines Test

Find the values of x and y. State which theorem(s) you used.

Question 1.

Question 2.

Question 3.

Find the distance from point A to the given line.

Question 4.
A(3, 4), y = – x

Question 5.
A(- 3, 7), y = $$\frac{1}{3}$$x – 2

Find the value of x that makes m || n.

Question 6.

Question 7.

Question 8.

Write an equation of the line that passes through the given point and is
(a) parallel to and
(b) perpendicular to the given line.

Question 9.
(- 5, 2), y = 2x – 3

Question 10.
(- 1, – 9), y = – $$\frac{1}{3}$$x + 4

Question 11.
A student says. “Because j ⊥ K, j ⊥ l’ What missing information is the student assuming from the diagram? Which theorem is the student trying to use?

Question 12.
You and your family are visiting some attractions while on vacation. You and our your mom visit the shopping mall while your dad and your sister visit the aquarium. You decide to meet at the intersection of lines q and p. Each unit in the coordinate plane corresponds to 50 yards.

a. Find an equation of line q.

b. Find an equation of line p.

c. What are the coordinates of the meeting point?

d. What is the distance from the meeting point to the subway?

Question 13.
Identify an example on the puzzle cube of each description. Explain your reasoning.

a. a pair of skew lines

b. a pair of perpendicular lines

c. a pair of partlIeI lines

d. a pair of congruent corresponding angles

e. a pair of congruent alternate interior angles

### Parallel and Perpendicular Lines Cumulative Assessment

Question 1.
Use the steps in the construction to explain how you know that im – 197 is the perpendicular bisector of $$\overline{A B}$$.

Question 2.
The equation of a line is x + 2y = 10.

a. Use the numbers and symbols to create the equation of a line in slope-intercept form
that passes through the point (4, – 5) and is parallel to the given line.

b. Use the numbers and symbols to create the equation of a line in slope-intercept form
that passes through the point (2, – 1) and is perpendicular to the given line.

Question 3.
Classify each pair of angles whose measurements are given.
a.

b.

c.

d.

Question 4.
Your school is installing new turf on the football held. A coordinate plane has been superimposed on a diagram of the football field where 1 unit = 20 feet.

a. What is the length of the field?

b. What is the perimeter of the field?

c. Turf costs $2.69 per square foot. Your school has a$1,50,000 budget. Does the school have enough money to purchase new turf for the entire field?

Question 5.
Enter a statement or reason in each blank to complete the two-column proof.

Given ∠1 ≅∠3
Prove ∠2 ≅∠4
Table – 1

Question 6.
Your friend claims that lines m and n are parallel. Do you support your friend’s claim? Explain your reasoning.

Question 7.
Which of the following is true when are skew?
(A) are parallel.
(B) intersect
(C) are perpendicular
(D) A, B, and C are noncollinear.

Question 8.
Select the angle that makes the statement true.

∠1    ∠2    ∠3    ∠4    ∠5     ∠6     ∠7     ∠8
a. ∠4 ≅ ________ b the Alternate Interior Angles Theorem (Thm. 3.2).

b. ∠2 ≅ ________ by the Corresponding Angles Theorem (Thm. 3. 1)

c. ∠1 ≅ ________ by the Alternate Exterior Angles Theorem (Thm. 3.3).

d. m∠6 + m ________ = 180° by the Consecutive Interior Angles Theorem (Thm. 3.4).