correctionkey=nl-b;ca-b lesson 19 . 4 do … when you draw a perpendicular line from p to the line...

© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Name Class Date

1 Explore Constructing Perpendicular Bisectors and Perpendicular Lines

You can construct geometric figures without using measurement tools like a ruler or a protractor. By using geometric relationships and a compass and a straightedge, you can construct geometric figures with greater precision than figures drawn with standard measurement tools.

In Steps A–C, construct the perpendicular bisector of AB .

A Place the point of the compass at point A. Using a compass setting that is greater than half the length of

_ AB , draw an arc.

B Without adjusting the compass, place the point of the compass at point B and draw an arc intersecting the first arc in two places. Label the points of intersection C and D.

C Use a straightedge to draw

_ CD , which is the

perpendicular bisector of _ AB .

In Steps D–E, construct a line perpendicular to a line ℓ that passes through some point P that is not on ℓ .

D Place the point of the compass at P. Draw an arc that intersects line ℓ at two points, A and B.

E Use the methods in Steps A–C to construct the perpendicular bisector of _ AB .

Because it is the perpendicular bisector of _ AB , then the constructed

line through P is perpendicular to line ℓ.

Resource Locker

A

C

D

B A

C

D

BA B

A B

P

ℓ

P

A Bℓ

A Bℓ P

Module 19 965 Lesson 4

19.4 Perpendicular LinesEssential Question: What are the key ideas about perpendicular bisectors of a segment?

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

IN1_MNLESE389762_U8M19L4 965 9/23/14 9:04 AM

Common Core Math StandardsThe student is expected to:

G-CO.9

Prove theorems about lines and angles. Also G-CO.12, MP.3.1

Mathematical Practices

MP.5 Using Tools

Language ObjectiveExplain to a partner why a pair of lines is or is not perpendicular.

HARDCOVER PAGES 777784

Turn to these pages to find this lesson in the hardcover student edition.

Perpendicular Lines

ENGAGE Essential Question: What are the key ideas about perpendicular bisectors of a segment?Possible answer: If you know that a line is the perpendicular bisector of a segment, then any point on the line is equidistant from the endpoints of the segment. If you know that a point on a line is equidistant from the endpoints of a segment, then the line must be the perpendicular bisector of the segment.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo. Ask students how they would balance the competing factors that a power company might face in deciding where to build a wind farm. Then preview the Lesson Performance Task. 965

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

© H

ough

ton

Mif

flin

Har

cour

t Pub

lishi

ng C

omp

any

Name

Class Date

1 Explore Constructing Perpendicular Bisectors

and Perpendicular Lines

You can construct geometric figures without using measurement tools like a ruler or a

protractor. By using geometric relationship with a compass and a straightedge, you can

construct geometric figures with greater precision than figures drawn with standard

measurement tools.

In Steps A–C, construct the perpendicular bisector of AB .

Place the point of the compass

at point A. Using a compass

setting that is greater than half

the length of _ AB , draw an arc.

Without adjusting the compass,

place the point of the compass at

point B and draw an arc intersecting

the first arc in two places. Label the

points of intersection C and D.

Use a straightedge to

draw _ CD , which is the

perpendicular bisector

of _ AB .

In Steps D–E, construct a line perpendicular to a line ℓ that passes through

some point P that is not on ℓ .

Place the point of the compass at P.

Draw an arc that intersects line ℓ at

two points, A and B.

Use the methods in Steps A–C to

construct the perpendicular bisector

of _ AB .

Because it is the perpendicular bisector of _ AB , then the constructed

line through P is perpendicular to line ℓ.

Resource

Locker

G-CO.9 Prove theorems about lines and angles. Also G-CO.12, MP.3.1

A

C

D

BA

C

D

B

AB

AB

P

ℓ

P

A Bℓ A B

ℓP

Module 19

965

Lesson 4

19 . 4 Perpendicular Lines

Essential Question: What are the key ideas about perpendicular bisectors of a segment?

DO NOT EDIT--Changes must be made through "File info"

CorrectionKey=NL-A;CA-A

IN1_MNLESE389762_U8M19L4 965

4/19/14 12:41 PM

965 Lesson 19 . 4

L E S S O N 19 . 4

DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-C;CA-C

© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Reflect

1. In Step A of the first construction, why do you open the compass to a setting that is greater than half the length of

_ AB ?

2. What If? Suppose Q is a point on line ℓ. Is the construction of a line perpendicular to ℓ through Q any different than constructing a perpendicular line through a point P not on the line, as in Steps D and E?

Explain 1 Proving the Perpendicular Bisector Theorem Using Reflections

You can use reflections and their properties to prove a theorem about perpendicular bisectors. These theorems will be useful in proofs later on.

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Example 1 Prove the Perpendicular Bisector Theorem.

Given: P is on the perpendicular bisector m of _ AB .

Prove: PA = PB

Consider the reflection across . Then the reflection

of point P across line m is also because point P lies

on , which is the line of reflection.

Also, the reflection of across line m is B by the definition

of .

Therefore, PA = PB because preserves distance.

Reflect

3. Discussion What conclusion can you make about △KLJ in the diagram using the Perpendicular Bisector Theorem?

A B

P

m

K M

J

L

P

line m

line m

point A

reflection

reflection

This ensures that the two arcs will intersect at two points.

Constructing the points A and B on line ℓ is different in the two constructions. For a

point Q on line ℓ, you place the compass point at Q and draw arcs on either side of Q. The

intersection points will be points A and B. Then, you can follow the same methods as in

Steps A–C in the Explore.

JK = JL because point J lies on the perpendicular

bisector of _ KL .


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-A;CA-A


IN1_MNLESE389762_U8M19L4 966 4/19/14 12:40 PM

Integrate Mathematical PracticesThis lesson provides an opportunity to address Mathematical Practice MP.5, which calls for students to “use appropriate tools.” They begin the lesson by constructing the perpendicular bisector of a segment; they may also use paper-folding or reflective devices to construct the perpendicular bisector. Students also construct the perpendicular to a line through a point not on the line. For each construction, they must be able to use the tools in a variety of ways in order to obtain accurate results and to understand the underlying mathematical relationships.

EXPLORE Constructing Perpendicular Bisectors and Perpendicular Lines

INTEGRATE TECHNOLOGYStudents have the option of exploring the perpendicular bisector activity either in the book or online.

QUESTIONING STRATEGIESWhy do you have to use a compass setting greater than half the length of the segment

when you construct the perpendicular bisector of the segment? This ensures that the two arcs will intersect.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Have students think about the steps used with each construction. Ask them to reflect on how they remember how to do each. In small groups, have students discuss strategies for remembering the steps. Then share each group’s best strategies with the class.

EXPLAIN 1 Proving the Perpendicular Bisector Theorem Using Reflections

QUESTIONING STRATEGIESSuppose you construct the perpendicular bisector of a segment and then choose any

point on the perpendicular bisector. If you measure the distance from the point to each endpoint of the segment, what do you expect to find? The distances from the point to each endpoint of the segment are equal.

PROFESSIONAL DEVELOPMENT

Perpendicular Lines 966


© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Your Turn

Use the diagram shown. _ BD is the perpendicular bisector of AC .

4. Suppose ED = 16 cm and DA = 20 cm. Find DC.

5. Suppose EC = 15 cm and BA = 25 cm. Find BC.

Explain 2 Proving the Converse of the Perpendicular Bisector Theorem

The converse of the Perpendicular Bisector Theorem is also true. In order to prove the converse, you will use an indirect proof and the Pythagorean Theorem.

In an indirect proof, you assume that the statement you are trying to prove is false. Then you use logic to lead to a contradiction of given information, a definition, a postulate, or a previously proven theorem. You can then conclude that the assumption was false and the original statement is true.

Recall that the Pythagorean Theorem states that for a right triangle with legs of length a and b and a hypotenuse of length c, a 2 + b 2 = c 2 .

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

Example 2 Prove the Converse of the Perpendicular Bisector Theorem

Given: PA = PB

Prove: P is on the perpendicular bisector m of _ AB .

Step A: Assume what you are trying to prove is false.

Assume that P is not on the perpendicular bisector m of . Then, when you draw a perpendicular line from P to the line containing A and B,

it intersects _ AB at point Q, which is not the of

_ AB .

Step B: Complete the following to show that this assumption leads to a contradiction.

_ PQ forms two right triangles, △AQP and .

So, A Q 2 + Q P 2 = P A 2 and B Q 2 + Q P 2 = by the Theorem.

Subtract these equations:

A Q 2 + QP 2 = P A 2

B Q 2 + Q P 2 = PB 2 ____

A Q 2 - B Q 2 = P A 2 - P B 2

However, P A 2 - P B 2 = 0 because .

Therefore, A Q 2 - B Q 2 = 0. This means that A Q 2 = B Q 2 and AQ = BQ. This contradicts the fact that Q is not the midpoint of

_ AB . Thus, the initial assumption must be incorrect,

and P must lie on the of _ AB .

a a2 + b2 = c2

b

c

A

B

C

D

E

m

A Q

P

B

Because BD is the perpendicular bisector of AC, then DA = DC and DC = 20 cm.

Because BD is the perpendicular bisector of AC, then BA = BC and BC = 25 cm.

PB 2

△BQP

Pythagorean

PA = PB

perpendicular bisector

AB

midpoint


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

IN1_MNLESE389762_U8M19L4 967 9/23/14 9:06 AM

COLLABORATIVE LEARNING

Whole Class ActivityHave groups of students create posters to describe how to do an indirect proof. Then demonstrate the method for the Converse of the Perpendicular Bisector Theorem. Ask them to display their results as a graphic organizer that shows the steps for the proof.

AVOID COMMON ERRORSWhen finding the distance from a point on one side of a perpendicular bisector to its reflected point, students may give the distance to the bisector as the solution. Remind students to re-read the question to verify that they have answered it by using given information, such as congruence markings.

EXPLAIN 2 Proving the Converse of the Perpendicular Bisector Theorem

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Make sure students understand that the proof of the Converse of the Perpendicular Bisector Theorem is based on the method of indirect proof. To write an indirect proof:

1. Identify the conjecture to be proven.

2. Assume the opposite of the conclusion is true.

3. Use direct reasoning to show the assumption leads to a contradiction.

4. Conclude that since the assumption is false, the original conjecture must be true.

QUESTIONING STRATEGIESHow can you tell that an indirect proof is used? In the first step, you assume that what

you are trying to prove is false.

Near the end of an indirect proof, a step contradicts a known true statement. What

does this mean in terms of the proof? The original assumption is false, so what you are trying to prove must be true.

967 Lesson 19 . 4


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Reflect

6. In the proof, once you know A Q 2 = B Q 2 , why can you conclude that AQ = BQ?

Your Turn

7. _ AD is 10 inches long.

_ BD is 6 inches long. Find the length of

_ AC .

Explain 3 Proving Theorems about Right AnglesThe symbol ⊥ means that two figures are perpendicular. For example, ℓ ⊥ m or ‹ −

› XY ⊥ _ AB .

Example 3 Prove each theorem about right angles.

If two lines intersect to form one right angle, then they are perpendicular and they intersect to form four right angles.

Given: m∠1 = 90° Prove: m∠2 = 90°, m∠3 = 90°, m∠4 = 90°

Statement Reason

1. m∠1 = 90° 1. Given

2. ∠1 and ∠2 are a linear pair. 2. Given

3. ∠1 and ∠2 are supplementary. 3. Linear Pair Theorem

4. m∠1 + m∠2 = 180° 4. Definition of supplementary angles

5. 90° + m∠2 = 180° 5. Substitution Property of Equality

6. m∠2 = 90° 6. Subtraction Property of Equality

7. m∠2 = m∠4 7. Vertical Angles Theorem

8. m∠4 = 90° 8. Substitution Property of Equality

9. m∠1 = m∠3 9. Vertical Angles Theorem

10. m∠3 = 90° 10. Substitution Property of Equality

If two intersecting lines form a linear pair of angles with equal measures, then the lines are perpendicular.

Given: m∠1 = m∠2 Prove: ℓ ⊥ m

By the diagram, ∠1 and ∠2 form a linear pair so ∠1 and ∠2 are supplementary

by the . By the definition of supplementary angles,

m∠1 + m∠2 = . It is also given that ,

so m∠1 + m∠1 = 180° by the . Adding

gives 2 ⋅ m∠1 = 180°, and m∠1 = 90° by the Division Property of Equality. Therefore,

∠1 is a right angle and ℓ ⊥ m by the .

A B

D

C

1 234

m

ℓ

1 2

Take the square root of both sides. Since distances are nonnegative, AQ = BQ.

Since D is equidistant from A and C and DB is perpendicular to AC by the diagram, then DB must be the perpendicular bisector of AC and AC = 2 · BC. B C 2 + 6 2 = 1 0 2 , so BC = 8 in. and AC = 16 in.

Linear Pair Theorem

180° m∠1 = m∠2

Substitution Property of Equality

definition of perpendicular lines




IN1_MNLESE389762_U8M19L4 968 9/23/14 9:09 AM

DIFFERENTIATE INSTRUCTION

Visual CuesUse colors to identify the steps in an indirect proof. Write each of the steps in a different color. Then place colored boxes around the parts of the proof that correspond to each step. For example, write the step “Assume the opposite of the conclusion is true” in blue and put a blue box around the assumption made in the indirect proof.

COMMUNICATING MATHHelp students understand the Converse of the Perpendicular Bisector Theorem by having them make a poster showing the theorem and its converse. Then have them explain what is given as the hypothesis of the theorem and of its converse, and what they have to prove for the theorem and its converse.

EXPLAIN 3 Proving Theorems About Right Angles

QUESTIONING STRATEGIESIf a linear pair of angles has equal measure, why are the angles right angles? Since a linear

pair of angles is supplementary, their sum must be 180°. So each angle must be half of 180°, or 90°.

INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 Encourage students to use the geometry software to create linear pairs of angles and then measure them. Then have them construct perpendicular lines and measure each of the angles to verify that they are right angles.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect

8. State the converse of the theorem in Part B. Is the converse true?

Your Turn

9. Given: b ǁ d, c ǁ e , m∠1 = 50°, and m∠5 = 90°. Use the diagram to find m∠4.

Elaborate

10. Discussion Explain how the converse of the Perpendicular Bisector Theorem justifies the compass-and-straightedge construction of the perpendicular bisector of a segment.

11. Essential Question Check-In How can you construct perpendicular lines and prove theorems about perpendicular bisectors?

a

b

c

d

e

1

2

34 5

6

A

C

D

B

If two intersecting lines are perpendicular, then they form a linear pair of angles with

equal measures; yes.

m∠4 = 40°; by corresponding angles because c ǁ e, m∠1 = m∠2, and by vertical angles, m∠2 = m∠3, so m∠3 = 50°; because m∠5 = 90°, then a ⊥ d and m∠3 + m∠4 = 90, so m∠4 = 40°.

The construction involves making two arcs that intersect in two points. Each of these two

intersection points is equidistant from the endpoints of the segment, because the arcs are

the same radius. So, both of the intersection points are on the perpendicular bisector of

the segment.

Constructing a line perpendicular to a given line involves using a compass to locate two

points that are not on the given line but are equidistant from two points on the given line.

You can prove the Perpendicular Bisector Theorem using a reflection and its properties,

and you can prove the Converse of the Perpendicular Bisector Theorem using an indirect

argument involving the Pythagorean Theorem.



IN1_MNLESE389762_U8M19L4 969 9/23/14 9:16 AM

LANGUAGE SUPPORT

Visual CuesHave students work in pairs. Give students pictures of intersecting, parallel, skew, and perpendicular lines. Instruct one student in each pair to explain why a pair of lines is or is not perpendicular, and prove it to the partner. Have the student who is not explaining write notes about the proof explanation. Then have students switch roles.

ELABORATE QUESTIONING STRATEGIES

How is constructing a perpendicular bisector related to constructing a segment

bisector? The construction of a perpendicular bisector starts with a segment.

How is constructing a perpendicular bisector related to constructing a perpendicular to a

point on a line? The construction of a perpendicular to a point on a line starts with marking equal distances from the point along the line. This creates a segment with the point as the midpoint of the segment.

SUMMARIZE THE LESSONIf you are given a line and a point P, how do you construct a line that is perpendicular to the given line using a compass and straightedge? Sample answer: Use a compass to locate two points A and B on the line that are the same distance from point P. Then construct the perpendicular bisector of

_ AB .

969 Lesson 19 . 4


• Online Homework• Hints and Help• Extra Practice

© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany • Im

age C

redits: ©

sima/

Shutterstock

1. How can you construct a line perpendicular to line ℓ that passes through point P using paper folding?

2. Check for Reasonableness How can you use a ruler and a protractor to check the construction in Elaborate Exercise 11?

3. Describe the point on the perpendicular bisector of a segment that is closest to the endpoints of the segment.

4. Represent Real-World Problems A field of soybeans is watered by a rotating irrigation system. The watering arm, _CD , rotates around its center point. To show the area of the crop of soybeans that will be watered, construct a circle with diameter CD.

Use the diagram to find the lengths. _ BP is the perpendicular bisector

of _ AC . _ CQ is the perpendicular bisector of

_ BD . AB = BC = CD.

5. Suppose AP = 5 cm. What is the length of _ PC ? 6. Suppose AP = 5 cm and BQ = 8 cm. What is the

length of _ QD ?

Evaluate: Homework and Practice

P

ℓ

C D

A B

P

Q

C D

Fold line ℓ onto itself so that the crease passes through point P.

The crease is the required perpendicular line.

Use the ruler to check that ‹ −

› CD bisects AB.

Use the protractor to check that AB and ‹ −

› CD

are perpendicular.

The midpoint of the segment is the point

on the perpendicular bisector that is

closest to the endpoints of the segment.

By the Perpendicular Bisector Theorem,

PC = 5 cm.

By the Perpendicular Bisector Theorem,

QD = 8 cm.




IN1_MNLESE389762_U8M19L4 970 4/19/14 12:40 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1 1 Recall of Information MP.5 Using Tools

2–4 2 Skills/Concepts MP.5 Using Tools

5–13 2 Skills/Concepts MP.6 Precision

14 3 Strategic Thinking MP.6 Precision

15–16 2 Skills/Concepts MP.6 Precision

17 3 Strategic Thinking MP.2 Reasoning

18 3 Strategic Thinking MP.3 Logic

19 3 Strategic Thinking MP.2 Reasoning

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreConstructing Perpendicular Bisectors and Perpendicular Lines

Exercises 1–4

Example 1Proving the Perpendicular Bisector Theorem

Exercises 5–8, 15–16

Example 2Proving the Converse of the Perpendicular Bisector Theorem

Exercises 9–11, 14

Example 3Proving Theorems about Right Angles

Exercises 12–13, 15–16

AVOID COMMON ERRORSA common error when working with perpendicular lines is to assume that lines are perpendicular if they look perpendicular. Emphasize the need to establish that there are right angles before saying that lines are perpendicular.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

7. Suppose AC = 12 cm and QD = 10 cm. What is the length of

_ QC ?

8. Suppose PB = 3 cm and AD = 12 cm . What is the length of

_ PC ?

Given: PA = PC and BA = BC. Use the diagram to find the lengths or angle measures described.

9. Suppose m∠2 = 38°. Find m∠1.

10. Suppose PA = 10 cm and PB = 6 cm. What is the length of

_ AC ?

11. Find m∠3 + m∠4.

Given: m ǁ n, x ǁ y, and y ⊥ m. Use the diagram to find the angle measures.

12. Suppose m∠7 = 30°. Find m∠3. 13. Suppose m∠1 = 90°. What is m∠2 + m∠3 + m∠5 + m∠6?

A

P

B C

1 234

x

m na

y

1234

5

78

6

AC = 16 cm; because PA = PC and BA = BC,

then ‹ −

› PB is the perpendicular bisector of AC and

△PBA is a right triangle. By the Pythagorean

Theorem, P A 2 = P B 2 + B A 2 , so 1 0 2 = 6 2 + B A 2

and BA = 8 cm. Then AC = 2 ⋅ BA = 16 cm.

m∠3 + m∠4 = 90°; because P and B are

equidistant from the endpoints, then ‹ −

› PB

is the perpendicular bisector of _ AC .

So, ‹ −

› PB ⊥ ‹ −

› AC and the lines meet at right

angles, and m∠3 + m∠4 = 90°.

QC = 8 cm; AC = 12 cm so BC = 1 _ 2

⋅

AC = 6 cm and CD = 6 cm. By the

Pythagorean Theorem, Q D 2 = Q C 2 + C D 2 ,

so 1 0 2 = Q C 2 + 6 2 and QC = 8 cm.

PC = 5 cm; AD = 12 cm, so AB = BC = CD = 4 cm.

By the Pythagorean Theorem, P C 2 = P B 2 + BC 2 ,

so P C 2 = 3 2 + 4 2 and PC = 5 cm.

m∠1 = 52°; because PA = PC and BA = BC, then ‹ −

› PB is

the perpendicular bisector of AC, and m∠PBC = 90°.

Then m∠1 = 90° - 38° = 52°.

y ⊥ m, so m∠7 + m∠8 = 90° and m∠8 = 60°.

Using alternate interior angles, because x ǁ y,

then m∠8 = m∠6, so m∠6 = 60°. ∠6 and ∠3

are vertical angles, so m∠3 = 60°.

Because m∠1 = 90°, then x ⊥ n and the

lines intersect at four right angles. So,

m∠2 + m∠3 + m∠5 + m∠6 = 180°.



IN1_MNLESE389762_U8M19L4 971 9/23/14 9:17 AM

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Ask students to compare constructions using a reflective device, tracing paper, compass and straightedge, and geometry software. Discuss the advantages of each. Have students consider not only how easy each is to use, but also how accurate, and how well each tool helps them understand the geometric relationships being studied.

INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 Ask each student to draw a simple sketch that involves perpendicular lines and right angles. Have students exchange sketches and attempt to reproduce the one they receive using geometry software.

971 Lesson 19 . 4


© H

oughton Mifflin H

arcourt Publishing Company

Use this diagram of trusses for a railroad bridge in Exercise 14.

14. Suppose _ BE is the perpendicular bisector of

_ DF . Which of the following statements do you

know are true? Select all that apply. Explain your reasoning.

A. BD = BF

B. m∠1 + m∠2 = 90°

C. E is the midpoint of _ DF .

D. m∠3 + m∠4 = 90°

E. _ DA ⊥ _ AC

15. Algebra Two lines intersect to form a linear pair with equal measures. One angle has the measure 2x° and the other angle has the measure (20y - 10) °. Find the values of x and y. Explain your reasoning.

16. Algebra Two lines intersect to form a linear pair of congruent angles. The measure

of one angle is (8x + 10) ° and the measure of the other angle is ( 15y ___ 2 ) °. Find the values

of x and y. Explain your reasoning.

H.O.T. Focus on Higher Order Thinking

17. Communicate Mathematical Ideas The valve pistons on a trumpet are all perpendicular to the lead pipe. Explain why the valve pistons must be parallel to each other.

D E F

C21

43

A B

valve pistons

lead pipelead pipe

valve pistonsvalve pistons

x = 45, y = 5; the linear pair formed has equal measures, so the lines are

perpendicular and then 2x° = 90°, so x = 45, and (20y - 10) ° = 90°, so y = 5.

x = 10, y = 12; the linear pair formed are congruent angles, so the lines are

perpendicular and then (8x + 10) ° = 90°, so x = 10, and ( 15y

___ 2 ) ° = 90°, so y = 12.

The valve pistons are lines that are perpendicular to the same line (the lead pipe), so they form right angles with the same line. By the converse of the corresponding angles theorem, all the congruent right angles mean the valve pistons are parallel to each other.

A, C, and D; the given information that BE is the perpendicular bisector

of DF means that both points E and B are equidistant from D and F so

answer choices A and C are true. Then, EB ⊥ DF and m∠3 + m∠4 = 90°,

so answer choice D is known to be true. The given information does not tell anything about

DA or AC, though, so answer choices B and E may or may not be true.


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-C;CA-C

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-C;CA-C

IN1_MNLESE389762_U8M19L4 972 09/06/15 12:04 PM

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Remind students about how to use the Perpendicular Bisector Theorem and its converse to write equations that will help them find angle measures related to perpendicular lines.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Have students explain how they can use a protractor or reflective device to check the accuracy of their constructions.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

18. Justify Reasoning Prove the theorem: In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Given: _ RS ⊥ _ CD and

_ AB ∥ _ CD Prove:

_ RS ⊥

_ AB

Statements Reasons

19. Analyze Mathematical Relationships Complete the indirect proof to show that two supplementary angles cannot both be obtuse angles.

Given: ∠1 and ∠2 are supplementary.

Prove: ∠1 and ∠2 cannot both be obtuse.

Assume that two supplementary angles can both be obtuse angles. So, assume that

∠1 and ∠2 . Then m∠1 > 90° and m∠2 >

by . Adding the two inequalities,

m∠1 + m∠2 > . However, by the definition of supplementary angles,

. So m∠1 + m∠2 > 180° contradicts the given information.

This means the assumption is , and therefore

.

R

DTC

SBVA

1. Given

2. Corresponding Angles Theorem

3. Given

4. Definition of perpendicular lines

5. Substitution Property of Equality

6. Definition of perpendicular lines

1. _ AB ǁ CD

2. m∠RTD = m∠RVB

3. RS ⊥ CD

4. m∠RTD = 90°

5. m∠RVB = 90°

6. RS ⊥ AB

m∠1 + m∠2 = 180°

∠1 and ∠2 cannot both be obtuse

are both obtuse 90°

180°

the definition of obtuse angles

false



IN1_MNLESE389762_U8M19L4 973 9/23/14 9:26 AM

COLLABORATIVE LEARNING ACTIVITYHave students work in groups of three or four. Ask them to choose one student to give directions. Instruct the other students to each draw a line and a point that is not on the line. The first student will then give a set of directions, step by step, for constructing a perpendicular to a line through a point that is either on the line or not on the line. The other students will not know which construction it is until they have followed the directions. Once the student has successfully guided the other students through the construction process, have another student take the lead and describe the construction to the others.

JOURNALHave students draw and mark a set of figures to illustrate the Perpendicular Bisector Theorem and its converse. Have students provide explanatory captions for the figures.

973 Lesson 19 . 4


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

A utility company wants to build a wind farm to provide electricity to the towns of Acton, Baxter, and Coleville. Because of concerns about noise from the turbines, the residents of all three towns do not want the wind farm built close to where they live. The company comes to an agreement with the residents to build the wind farm at a location that is equally distant from all three towns.

a. Use the drawing to draw a diagram of the locations of the towns using a scale of 1 in. : 10 mi. Draw the 4-inch and 1.5-inch lines with a 120° angle between them. Write the actual distances between the towns on your diagram.

b. Estimate where you think the wind farm will be located.

c. Use what you have learned in this lesson to find the exact location of the wind farm. What is the approximate distance from the wind farm to each of the three towns?

Lesson Performance Task

1.5 in.

4 in.

Acton

Coleville BaxterScale 1 in. : 10 mi

120°

a. Distances: Acton–Baxter, about 49 miles; Baxter–Coleville, 40 miles; Acton–Coleville, 15 miles

b. Student answers will vary.

c. Students should construct perpendicular bisectors of the three lines of the triangle. The point of intersection of the bisectors is the point that is equidistant from the three vertices of the triangle. Any two of the bisectors is sufficient to locate the point, but a third one is useful for spotting possible errors in the drawing of the first two bisectors.


DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A

IN1_MNLESE389762_U8M19L4 974 4/19/14 12:40 PM

EXTENSION ACTIVITY

Pose this challenge to students: Three other towns have signed up to obtain electricity from the wind farm. All are the same distance from the farm as Acton, Baxter, and Coleville. On your drawing, show three points that could be the locations of the towns. Explain how you found the points and how you know they meet the conditions of the problem.

Students should draw a circle with its center at the wind farm and passing through Acton, Baxter, and Coleville. They can choose any three points on the circle as the locations of the three new towns. Those points must be the same distance from the farm as Acton, Baxter, and Coleville because all radii of a circle are equal in length.

AVOID COMMON ERRORSStudents can use proportions to find the actual distances between towns, but they may set them up incorrectly. To find the distance between Acton and Coleville, they can write and solve this proportion: 1 in. _____ 10 mi = 1.5 in. ______ x mi . The correct pattern is

map distance _____________ actual distance =

map distance _____________ actual distance .

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Marcus said that it isn’t necessary to draw three perpendicular bisectors to find the location of the wind farm. Two is sufficient, he said. Was he right? Explain. Yes; sample answer: Any two of the perpendicular bisectors will intersect at a point. That point must be the location of the wind farm because the two lines can intersect in only one point. The third perpendicular bisector provides a useful check on the accuracy of the constructions of the first two perpendicular bisectors.

Scoring Rubric

2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



correctionkey=nl-b;ca-b lesson 19 . 4 do … when you draw a perpendicular line from p to the line...

Documents