Common Core State Standards 9-5 G-CO.B.7 Transformations

Problem 1

578 Chapter 9 Transformations

Objective To identify congruence transformations To prove triangle congruence using isometries

Congruence Transformations

9-5

Suppose that you want to create two identical wings for a model airplane. You draw one wing on a large sheet of tracing paper, fold it along the dashed line, and then trace the first wing. How do you know that the two wings are identical?

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.

Are there other methods you could use to create two identical wings?

Lesson Vocabulary

•congruent•congruence

transformation

LessonVocabulary

Identifying Equal Measures

The composition (Rn ∘ r(90°, P))(LMNO) = GHJK 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 ∘ T62, 37)(△ABC) = △XYZ . List all of the pairs of angles and sides with equal measures.

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Got It?

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.

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

MP 1, MP 3, MP 4

MATHEMATICAL PRACTICES

Common Core State Standards

Problem 2

Lesson 9-5 CongruenceTransformations 579

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 (T6-1, 57 ∘ Ry@axis)(ABCJ) = WXYZ , the trapezoids are congruent.

Because T6-2, 97(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.

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

Problem 3


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.

T60, 57(x, y) = (x, y + 5) T60, 57(J) = (2, 4)

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

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

T60, 57(Q) = (1, 1) T60, 57(V) = (5, 2) Ry@axis(1, 1) = (-1, 1) = W Ry@axis(5, 2) = (-5, 2) = T

So, the congruence transformation Ry@axis ∘ T60, 57 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 ?


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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


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Problem 4


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 △PAF 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 and 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 △JOS, so △PAF ≅ △JOS.

4. Verify the SSS postulate.

Given: TD ≅ EN , YT ≅ SE, YD ≅ SN

Prove: △YDT ≅ △SNE

Proof


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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.

Problem 5


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

Got It?


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?


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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.


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




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.



See Problem 3.


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See Problem 4.Proof


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See Problem 5.

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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 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 .

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Proof


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) S (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


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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

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Common Core State Standards 9-5 G-CO.B.7 Transformations