Problem 1

Got It?

Objective To identify congruence transformations

To prove triangle congruence using isometries

Congruence Transformations

9-5

In the Solve It, you may have used the properties of rigid motions to describe why the

wings are identical.

Essential Understanding You can use compositions of rigid motions to

understand congruence.

Lesson Vocabularycongruent

transformation

LV

Identifying Equal Measures

The composition (r(9 0 , P) Rn)(LMNO) GHJ K is shown at the right.

A Which angle pairs have equal measures?

Because compositions of isometries preserve angle measure,

corresponding angles have equal measures.

m L m G, m M m H , m N m J , and

m O m K .

B Which sides have equal lengths?

By definition, isometries preserve distance. So,

corresponding side lengths have equal measures.

LM GH, MN HJ, NO JK, and LO GK.

1. The composition (Rt T 2, 3 )( ABC) XYZ. List all of the pairs of

angles and sides with equal measures.

n

P HG

J

K

L M

NO

How can you use the properties of isometries to find equal angle measures and equal side lengths? Isometries preserve angle measure and distance, so identify corresponding angles and corresponding side lengths.

Content StandardsG.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent . . .Also G.CO.6, G.CO.8

MATHEMATICAL PRACTICES

1 Common Core

CC-20CC-14

Problem 2

Got It?

In Problem 1 you saw that compositions of rigid motions preserve corresponding side

lengths and angle measures. This suggests another way to define congruence.

Identifying Congruent Figures

Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that maps one figure to the other?

Figures are congruent if and only if there is a sequence

of rigid motions that maps one figure to the other. So, to

find congruent figures, look for sequences of translations,

rotations, and reflections that map one figure to another.

Because r(180 , O)( DEF) LMN , the triangles are

congruent.

Because (T 1, 5 Ry-axis)(ABCJ) WXYZ, the

trapezoids are congruent.

Because T 2, 9 (HG) PQ, the line segments are

congruent.

2. Which pairs of figures in the grid are congruent? For each pair, what is a

sequence of rigid motions that map one figure to the other?

Key Concept Congruent Figures

Two figures are congruent if and only if there is a sequence of one or more rigid motions that

maps one figure onto the other.

y

xO2 2

2

6 6

6

4

2

6

4

4

HG

D

E

F

A B

CJ

L

M

N

Y

X WQ

P

Z

y

xO 2

2

6 6

6

4

2

6

4

4

4

F

B

A

C

D UV

W

H K

I JQ

N

M

Does one rigid motion count as a sequence? Yes. It is a sequence of length 1.

CC-14 Congruence Transformations 2

Problem 3

Got It?

Because compositions of rigid motions take figures to congruent figures, they are also

called congruence transformations.

Identifying Congruence Transformations

In the diagram at the right, JQV EWT. What is a congruence transformation that maps JQV onto EWT ?

Because EWT lies above JQV on the plane, a translation

can map JQV up on the plane. Also, notice that EWT is on the

opposite side of the y-axis and has the opposite orientation of

JQV. This suggests that the triangle is reflected across the y-axis.

It appears that a translation of JQV up 5 units, followed by a

reflection across the y-axis maps JQV to EWT. Verify by

using the coordinates of the vertices.

T 5, 0 (x, y) (x 5, y)

T 5, 0 (J) (2, 4)

Ry-axis(2, 4) ( 2, 4) E

Next, verify that the sequence maps Q to W and V to T.

T 5, 0 (Q) (1, 1) T 5, 0 (V) (5, 2)

Ry-axis(1, 1) ( 1, 1) W Ry-axis(5, 2) ( 5, 2) T

So, the congruence transformation Ry-axis T 5, 0 maps JQV onto EWT . Note that

there are other possible congruence transformations that map JQV onto EWT .

3. What is a congruence transformation that maps

NAV to BCY ?

y

x

T

E

Q

J

V

WO

2

4

2

4

42

4

Identify the corresponding parts and find a congruence transformation that maps the preimage to the image. Then use the vertices to verify the congruence transformation.

A sequence of rigid motions that maps JQV onto EWT

The coordinates of the vertices of the triangles

y

x

T

E

Q

J

V

WO

2

4 2

2

4

4

4

y

x

A

C

Y

B

N

VO 22

2

4

4

4

4

3 Common Core

Problem 4

Got It?

In Chapter 4, you studied triangle congruence postulates and theorems. You

can use congruence transformations to justify criteria for determining triangle

congruence.

Verifying the SAS Postulate

Given: J P, PA JO, FP SJ

Prove: JOS PAF

Step 1 Translate PAF so that points A and O coincide.

Step 2 Because PA JO, you can rotate JOS about point A

so that PA and JO coincide.

Step 3 Reflect PAF across PA. Because reflections preserve

angle measure and distance, and because J P

because FP SJ , you know that the reflection maps

P to J and FP to SJ . Since points S and F coincide,

PAF coincides with JOS.

There is a congruence transformation that maps PAF onto

SOJ, so PAF JOS.

4. Verify the SSS postulate.

Given: TD EN , YT SE , YD SN

Prove: YDT SNE

Proof

S

O

J

P

F A

O

S

J

A

P

F

P

A

F

O

J

S

OA

FS

PJ

T

D

E

S

N

Y

In Problem 4, you used the transformational approach to prove triangle congruence.

Because this approach is more general, you can use what you know about congruence

transformations to determine whether any two figures are congruent.

How do you show that the two triangles are congruent?Find a congruence transformation that maps one onto the other.

CC-14 Congruence Transformations 4

Problem 5

Got It?

Determining Congruence

Is Figure A congruent to Figure B? Explain how you know.

Figure A can be mapped to Figure B by a sequence of

reflections or a simple translation. So, Figure A is congruent

to Figure B because there is a congruence transformation

that maps one to the other.

5. Are the figures shown at the right congruent?

Explain.

Figure A

Figure B

Do you know HOW?Use the graph for Exercises 1 and 2.

1. Identify a pair of

congruent figures and

write a congruence

statement.

2. What is a congruence

transformation that

relates two congruent

figures?

Do you UNDERSTAND? 3. How can the definition of congruence in terms of

rigid motions be more useful than a definition of

congruence that relies on corresponding angles

and sides?

4. Reasoning Is a composition of a rotation followed by a

glide reflection a congruence transformation? Explain.

5. Open Ended What is an example of a board game in

which a game piece is moved by using a congruence

transformation?

R

x

y

V

A

T B

K

QS

I

O

2

6 4 2

2

6

4

4

Lesson Check

Practice and Problem-Solving Exercises

For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.

6. 7. 8.

PracticeA See Problem 1 and 2.

x

y

B

J

T

V

Q

E

YL

G

2 44

4

4

G

A

D

F

C

R

y

xO4 2

2

4

4

4x

y

FA

E

K

S

T

B

D

M

I

C

2

4

4

4

How can you determine whether the figures are congruent?You can find a congruence transformation that maps Figure A onto Figure B.

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

5 Common Core

In Exercises 9–11, find a congruence transformation that maps LMN to RST .

9. 10. 11.

12. Verify the ASA Postulate for triangle congruence by using congruence

transformations.

Given: EK LH Prove: EKS HLA

E H

K L

13. Verify the AAS Postulate for triangle congruence by using congruence

transformations.

Given: I V Prove: NVZ CIQ

C N

QC NZ

In Exercises 14–16, determine whether the figures are congruent. If so, describe a congruence transformation that maps one to the other. If not, explain.

14. 15. 16.

Construction The figure at the right shows a roof truss of a new building. Identify an isometry or composition of isometries to justify each of the following statements.

17. Triangle 1 is congruent to triangle 3.

18. Triangle 1 is congruent to triangle 4.

19. Triangle 2 is congruent to triangle 5.

See Problem 3.

x

y

L

M N

S

R

T

O 2

2

4

4

x

yL

M

N

SR

TO 2 4

2

4 2

4

x

y

L

M

N

S

T

RO 2 4

2

4

4

See Problem 4.Proof

L

H

A

E K

S

Proof

ZV

N

C

I Q

See Problem 5.

ApplyB

1

2 5

3 4

CC-14 Congruence Transformations 6

20. Vocabulary If two figures are ________________, then there is an isometry that

maps one figure onto the other.

21. Think About a Plan The figure at the right shows two congruent,

isosceles triangles. What are four different isometries that map

the top triangle onto the bottom triangle?

How can you use the three basic rigid motions to map the top

triangle onto the bottom triangle?

What other isometries can you use?

22. Graphic Design Most companies have a logo that

is used on company letterhead and signs. A graphic

designer sketches the logo at the right. What congruence

transformations might she have used to draw this logo?

23. Art Artists frequently use congruence transformations in their work. The artworks

shown below are called tessellations. What types of congruence transformations

can you identify in the tessellations?

a. b.

24. In the footprints shown below, what congruence transformations can you use to

extend the footsteps?

25. Prove the statements in parts (a) and (b) to show congruence in terms of

transformations is equivalent to the criteria of for triangle congruence you learned

in Chapter 4.

a. If there is a congruence transformation that maps ABC to DEF then

corresponding pairs of sides and corresponding pairs of angles are congruent.

b. In ABC and DEF , if corresponding pairs of sides and corresponding pairs

of angles are congruent, then there is a congruence transformation that maps

ABC to DEF .

x

y

O 2 44 2

Proof

7 Common Core

Mixed Review 33. A triangle has vertices A(3, 2), B(4, 1), and C(4, 3). Find the coordinates

of the images of A, B, and C for a glide reflection with translation

(x, y) (x, y 1) and reflection line x 0.

The lengths of two sides of a triangle are given. What are the possible lengths for the third side?

34. 16 in., 26 in. 35. 19.5 ft, 20.5 ft 36. 9 m, 9 m 37. 412

yd, 8 yd

Get Ready! To prepare for Lessons 9-6, do Exercises 38–40.

Determine the scale drawing dimensions of each room using a scale of 14 in. 1 ft.

38. kitchen: 12 ft by 16 ft 39. bedroom: 8 ft by 10 ft 40. laundry room: 6 ft by 9 ft

See Lesson 9-4.

See Lesson 5-6.

See Lesson 7-2.

26. Baking Cookie makers often use a cookie press so that

the cookies all look the same. The baker fills a cookie

sheet for baking in the pattern shown. What types of

congruence transformations are being used to set each

cookie on the sheet?

27. Use congruence transformations to prove the Isosceles Triangle Theorem.

Given: FG FH

Prove: G H

28. Reasoning You project an image for viewing in a large classroom. Is the projection of

the image an example of a congruence transformation? Explain your reasoning.

Proof

F

H

G

ChallengeC

Standardized Test Prep

29. To the nearest hundredth, what is the value of x in the diagram at the right?

30. In FGH and XYZ , G and Y are right angles. FH XZ and GH YZ .

If GH 7 ft and XY 9 ft, what is the area of FGH in square inches?

31. ACB is isosceles with base AB. Point D is on AB and CD is the bisector of C .

If CD 5 in. and DB 4 in., what is BC to the nearest tenth of an inch?

32. Two angle measures of JKL are 30 and 60. The shortest side measures 10 cm.

What is the length, in centimeters, of the longest side of the triangle?

SAT/ACT

20

xx

CC-14 Congruence Transformations 8