correctionkey=nl-b;ca-b lesson 5 . 1 do not edit--changes

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Explore Transforming Triangles with Congruent Corresponding Parts

You can apply what you’ve learned about corresponding parts of congruent figures to write the following true statement about triangles.

If two triangles are congruent, then the corresponding parts of the triangles are congruent.

The statement is sometimes referred to as CPCTC. The converse of CPCTC can be stated as follows.

If all corresponding parts of two triangles are congruent, then the triangles are congruent.

Use a straightedge and tracing paper to explore this converse statement.

A Trace the angles and segments shown to draw △ABC. Repeat the process to draw △DEF on a separate piece of tracing paper. Label the triangles.

B What must you do to show that the triangles are congruent?

C Flip the piece of tracing paper with △ABC and arrange the two triangles on a desk as shown in the figure. Then move the tracing paper with △ABC so that point A maps to point D. Name the rigid motion that you used.

Resource Locker

AB and DE

AC and DF

BC and EF∠C and ∠F

∠B and ∠E∠A and ∠D

DE

F

B

C

A

Find a sequence of rigid motions that maps one triangle onto the other.

translation

Module 5 219 Lesson 1

5.1 Exploring What Makes Triangles Congruent

Essential Question: How can you show that two triangles are congruent?

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GE_MNLESE385795_U2M05L1.indd 219 5/22/14 4:25 PM

Common Core Math StandardsThe student is expected to:

G-CO.B.7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Mathematical Practices

MP.2 Reasoning

Language ObjectiveHave students explain to a partner why a pair of triangles is congruent or noncongruent.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 191200

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

ENGAGE Essential Question: How can you show that two triangles are congruent?Show that their corresponding sides and

corresponding angles are congruent.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Explain that pollination is the process in which bees collect pollen from flowers and deposit it on other flowers. Then preview the Lesson Performance Task.

Exploring What Makes Triangles Congruent

219

HARDCOVER

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

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Explore Transforming Triangles with

Congruent Corresponding Parts

You can apply what you’ve learned about corresponding parts of congruent figures to write the

following true statement about triangles.

If two triangles are congruent, then the corresponding parts of the triangles are congruent.

The statement is sometimes referred to as CPCTC. The converse of CPCTC can be stated as follows.

If all corresponding parts of two triangles are congruent, then the triangles are congruent.

Use a straightedge and tracing paper to explore this converse statement.

Trace the angles and segments shown to draw △ABC. Repeat the process to draw △DEF on

a separate piece of tracing paper. Label the triangles.

What must you do to show that the triangles are congruent?

Flip the piece of tracing paper with △ABC and arrange the two triangles on a desk as shown

in the figure. Then move the tracing paper with △ABC so that point A maps to point D.

Name the rigid motion that you used.

Resource

Locker

G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles

are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are

congruent.

COMMONCORE

AB and DE

AC and DF

BC and EF

∠C and ∠F

∠B and ∠E

∠A and ∠D

D

E

F

B

C

A

Find a sequence of rigid motions that maps one triangle onto the other.

translation

Module 5

219

Lesson 1

5 . 1 Exploring What Makes

Triangles Congruent

Essential Question: How can you show that two triangles are congruent?

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GE_MNLESE385795_U2M05L1.indd 219

5/22/14 4:24 PM

219 Lesson 5 . 1

L E S S O N 5 . 1

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D Name a rigid motion you can use to map point B to point E. How can you be sure the image of B is E?

E Name a rigid motion you can use to map point C to point F.

F To show that the image of point C is point F, complete the following.

∠A is reflected across ‹ −

› DE , so the measure of the angle is preserved. Since ∠A ≅ ∠D, you

can conclude that the image of ‾→ AC lies on . It is given that

_ AC ≅ , so the

image of point C must be .

G What sequence of rigid motions maps △ABC onto △DEF?

Reflect

1. Discussion Is there another sequence of rigid motions that maps △ABC onto △DEF? Explain.

2. Discussion Is the converse of CPCTC always true when you apply it to triangles? Explain why or why not based on the results of the Explore.

D E

F

B

C

A

D E

F

B

C

A

Counterclockwise rotation about point D by m ∠BDE; _ AB ≅

_ DE , so the image of point B

must be point E.

reflection across ‹ −

› DE

a translation followed by a rotation followed by a reflection

Yes; possible answer: you can first rotate △ABC so that ‹ −

› AB || ‹ − › DE , then use a translation to

map A to D followed by a reflection across ‹ −

› DE to map △ABC onto △DEF.

Possible answer: Yes. If you draw two triangles that have congruent corresponding parts,

you will always be able to find a series of rigid motions that maps one onto the other.

So, the two triangles will always be congruent.

‾ → DF

point F

_ DF





EXPLORE Transforming Triangles with Congruent Corresponding Parts

INTEGRATE TECHNOLOGYHave students use geometry software to create designs using congruent triangles. They should arrange multiple congruent triangles using different colors, positions, and orientations. Ask them to make three separate designs, one using congruent equilateral triangles, one using congruent isosceles triangles, and one using congruent scalene triangles.

QUESTIONING STRATEGIESWhen you translate a triangle so that one vertex maps to a new location, what happens

to the other vertices and sides of the triangle? How does the triangle change? They are translated the

same direction and the same distance. The triangle

will have the same shape and size, but it will be in a

different position.

PROFESSIONAL DEVELOPMENT

Learning ProgressionsIn this lesson, students apply the rigid-motion definition of congruence specifically to triangles. Some key understandings for students are as follows:

• If two triangles are congruent, then all pairs of corresponding sides and all pairs of corresponding angles are congruent.

• If all pairs of corresponding sides and all pairs of corresponding angles of two triangles are congruent, then the triangles are congruent.

Students will explore ways to prove that triangles are congruent and use information gained from congruent triangles to solve problems.

Exploring What Makes Triangles Congruent 220


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Explain 1 Deciding If Triangles are Congruent by Comparing Corresponding Parts

A biconditional is a statement that can be written in the form “p if and only if q.” You can combine what you learned in the Explore with the fact that corresponding parts of congruent triangles are congruent to write the following true biconditional.

Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

To decide whether two triangles are congruent, you can compare the corresponding parts. If they are congruent, the triangles are congruent. If any of the corresponding parts are not congruent, then the triangles are not congruent.

Example 1 Determine whether the given triangles are congruent. Explain.

Compare corresponding sides to decide if they are congruent.

GH = KL = 1.7 m, HJ = LM = 1 m, and GJ = KM = 2 m.

So, _ GH ≅

_ KL , _ HJ ≅ _ LM , and

_ GJ ≅ _ KM .

Compare corresponding angles to decide if they are congruent.

m∠G = m∠K = 30°, m∠H = m∠L = 90°, and m∠J = m ∠M = 60°.

So, ∠G ≅ ∠K, ∠H ≅ ∠L, and ∠J ≅ ∠M.

△GHJ ≅ △KLM because all pairs of corresponding parts are congruent.

Compare corresponding sides to decide if they are congruent.

AB = = cm, so _ AB ≅ . AC = = cm, so

_ AC ≅ .

However, BC ≠ , so _ BC is not congruent to .

The triangles are not congruent because

G

H K

L M

J 1 m

1 m

2 m 2 m1.7 m 1.7 m

60°

60°

30°

30°

E D

F

12.1 cm

7.9 cm8.4 cm

A B

C

8.5 cm7.9 cm

12.1 cm

there is a pair of corresponding sides that are not

congruent.

DE DF12.1 DE DF

EF

7.9

EF


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COLLABORATIVE LEARNING

Peer-to-Peer ActivityWorking in pairs, one student draws and labels the vertices of a parallelogram and a rectangle that is not a square; the other draws a square and a trapezoid with exactly one pair of parallel sides. They exchange drawings and try to divide each quadrilateral into two congruent triangles by drawing one diagonal. For each divided quadrilateral, they identify the triangles’ corresponding sides and angles, write congruence statements, and review each other’s work.

EXPLAIN 1 Deciding if Triangles are Congruent by Comparing Corresponding Parts

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Make sure students pay attention to the order of the vertices when writing a congruence statement. The order of the vertices communicates which pairs of angles and sides are congruent. Have students write several correct congruence statements for the diagram. Then have them write an incorrect congruence statement and explain why it is not correct.

QUESTIONING STRATEGIESHow can you find the measure of a missing angle in a triangle with two given angle

measures? You know that the sum of the measures

of the angles of a triangle is always 180°, and you

know the other two angle measures in the triangle.

You can use this information to write and solve

an equation.

LANGUAGE SUPPORT Have students work in pairs to complete sentence stems explaining why a pair of triangles is congruent or not. For example: “This pair of triangles is congruent because ____________.” Possible

answers: The corresponding angles have equal

measures and the corresponding sides have equal

lengths; when I trace one I can fit it exactly over

the other.

221 Lesson 5 . 1


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Reflect

3. Critique Reasoning The contrapositive of a conditional statement “if p, then q” is the statement “If not q, then not p.” The contrapositive of a true statement is always true. Janelle says that you can justify Part B using the contrapositive of CPCTC. Is this accurate? Explain your reasoning.

Your Turn

Determine whether the given triangles are congruent. Explain your reasoning.

4.

5.

Explain 2 Applying Properties of Congruent TrianglesTriangles are part of many interesting designs. You can ensure that triangles are congruent by making corresponding sides congruent and corresponding angles congruent. To do this, you may have to use the Triangle Sum Theorem, which states that the sum of the measures of the angles of a triangle is 180°. You will explore this theorem in more detail later in this course.

71°

71°

71°

38°

71° 38°R S

U

V WT

41°

79°

60°

42°

78°

60°B C

EF

G

D

Yes; the contrapositive of a true statement is true, so the contrapositive of CPCTC is true:

if the corresponding parts of two figures are not congruent, then the figures are not

congruent. In Part B, the fact that at least one pair of corresponding parts is not congruent

means that the triangles are not congruent.

RS ≅ UV, ST ≅ VW, and RT ≅ UW. m∠R = m∠U = 71°, m∠S = m∠V = 38°, and m∠T = m∠W = 71°, so ∠R ≅ ∠U, ∠S ≅ ∠V and ∠T ≅ ∠W. △RST ≅ △UVW, because all six pairs of corresponding parts are congruent.

∠C ≅ ∠F, but m∠B ≠ m∠E and m∠D ≠ m∠G, so ∠B is not congruent to ∠E and ∠D is not congruent to ∠G. The triangles are not congruent because some pairs of the corresponding angles are not congruent.





DIFFERENTIATE INSTRUCTION

Critical ThinkingAsk students to use a ruler and protractor to draw and label two isosceles triangles ∆QRS and ∆UVW : m∠R = m∠V = 40°, m∠Q = m∠S = m∠U = m∠W = 70°, _

QS ≅ _

UW , and _

RQ ≅ _

RS ≅ _

VU ≅ _

VW . Then ask them to write a congruence statement for the triangles. It is likely that not all students will write the same congruence statement. Discuss why both ∆QRS ≅ ∆UVW and ∆QRS ≅ ∆WVU are correct even though the first maps point S onto point W and the second maps point S onto point U.

EXPLAIN 2 Applying Properties of Congruent Triangles

AVOID COMMON ERRORSStudents may correctly solve for a variable but then incorrectly give the value of the variable as a side length or angle measure. Remind them to examine the diagram carefully; sometimes a side length or angle measure is described by an expression containing a variable, not just the variable itself.

QUESTIONING STRATEGIESWhat must be true of the angle measures for the triangles to be congruent? The measures

of corresponding angles must be equal.

CONNECT VOCABULARY Have students review the prefixes bi-, tri-, quad-, and pent- to relate them to the number of sides and angles in a geometric figure.



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Example 2 Find the value of the variable that results in congruent triangles.

Step 1 Identify corresponding angles.

∠M corresponds to ∠J, because they have the same measure and they are formed by congruent corresponding sides. Similarly, ∠N corresponds to ∠K. So, ∠P corresponds to ∠L.

Step 2 Find m∠L.

Triangle Sum Theorem m∠J + m∠K + m∠L = 180°

Substitute. 55° + 45° + m∠L = 180°

Simplify. 100° + m∠L = 180°

Subtract 100° from each side. m∠L = 80°

Step 3 Write an equation to find the value of x.

Set corresponding measures equal. m∠P = m∠L

Substitute. 5x + 30 = 80

Subtract 30 from each side. 5x = 50

Divide each side by 5. x = 10

Step 1 Identify corresponding sides, beginning with side _ DE .

∠A ≅ ∠ , ∠B ≅ ∠ , and , ∠C ≅ ∠ , so _ DE corresponds to .

Step 2 Write an equation to find the value of y.

Set corresponding measures equal. DE = mm

Substitute. 2y + 20 =

Subtract 20 from each side. 2y =

Divide each side by 2. y =

55°

55°45°

45°

J

K

L

M

N

P(5x + 30)°

50 mm

30 mm36 mm

A

B

C

50 mm

(2y + 20) mm 30 mm

D F

E

D E F _ AB

36

36

16

8


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LANGUAGE SUPPORT

Connect ContextMany English words have multiple meanings. In this lesson, the word converse means opposite. The word is pronounced with the stress on the first syllable: “CON-verse.” English Learners may know the word as a verb, pronounced “con-VERSE,” meaning to have a conversation. Words in English typically do not have accent marks, so words that serve as both nouns and verbs can be difficult to distinguish. Remind students that when converse is preceded by the word to, it is a verb. When preceded by the word the, it is a noun or adjective.

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Reflect

6. The measures of two angles of △QRS are 18° and 84°. The measures of two angles of △TUV are 18° and 76°. Is it possible for the triangles to be congruent? Explain.

Your Turn

Find the value of the variable that results in congruent triangles.

7. 8.

Elaborate

9. All three angles of △ABC measure 60° and all three sides are 4 inches long. All three angles of △PQR measure 60° and all three sides are 4 inches long. Can you conclude that the triangles are congruent? Why or why not?

10. Use the concept of rigid motion to explain why two triangles cannot be congruent if any pair of corresponding parts is not congruent.

11. Essential Question Check-In △PQR and △STU have six pairs of congruent corresponding parts and △PQR can be mapped onto △STU by a translation followed by a rotation. How are the triangles related? Explain your reasoning.

(x + 17)°

B

CD

E

FG

112°

112°

25°

25°(4y + 12) mm

15 mm15 mm

32 mm22 mm

22 mm

J

K

L

M

NP

No; by the Triangle Sum Theorem, the measures of the angles of △QRS are 18°, 84°, and

78°, and the measures of the angles of △TUV are 18°, 76°, and 86°. Since two pairs of

corresponding angles are not congruent, the triangles cannot be congruent.

Yes; no matter how you set up the correspondence, corresponding angles will be

congruent and corresponding sides will be congruent. By the converse of CPCTC,

△ABC ≅ △PQR.

Possible answer: For figures to be congruent there must exist a sequence of rigid motions

that maps one to the other. If a pair of corresponding parts is not congruent, no rigid

motion maps one to the other because distance, or angle measure, will not be preserved.

The triangles are congruent by the converse of CPCTC and because a sequence of rigid

motions maps one triangle onto the other.

∠E corresponds to ∠B.

m∠B + 112° + 25° = 180°

m∠B = 43°

m∠E = 43°, so x + 17= 43, or x = 26

_ JK corresponds to

_ MN .

JK = 32 mm

4y + 12 = 32, so 4y = 20, or y = 5


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GE_MNLESE385795_U2M05L1.indd 224 3/20/14 9:51 AM

ELABORATE INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Have students explain how they could use transformations to decide whether the triangles are congruent.

SUMMARIZE THE LESSONSuppose that ∆BUD ≅ ∆RIN. Write six congruency statements. ∠B ≅ ∠R, ∠U ≅ ∠I ,

∠D ≅ ∠N, _

BU ≅ _

RI , _

UD ≅ _

IN , _

DB ≅ _

NR



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• Online Homework• Hints and Help• Extra Practice

1. Describe a sequence of rigid motions that maps △MNP onto △MQR to show that △MNP ≅ △MQR.

Determine whether the given triangles are congruent. Explain your reasoning.

2.

4.

3.

5.

Find the value of the variable that results in congruent triangles.

6. 7.

Evaluate: Homework and Practice

MR

Q N

P

32° 91°

57° 57°

91° 32°

K

L

Q

R SM

6.8 ft

6.8 ft

2.5 ft

2.5 ft 70º

70º

20º

20º

7.2 ft

7.2 ftA

BC

D

EF

7 in.

7 in.

R S

T U

V

W

35°

82°63°

35°

65°

80°32 ft

32 ft32 ft

32 ft

17 ft

19 ft

K

J

L

M

N

P

2.2 cm

3.1 cm

(3z - 3) cm

3.1 cm

1.8 cm

M

N

P

R

S

TA

B

C

F

G

H

69° 69°

50°

50°

(2x + 5)°

Possible answer: counterclockwise rotation about point M by m∠QMN followed by a reflection across

‹ − › MQ

― KL ≅ ― QR ,

― LM ≅ ― RS , ― KM ≅ QS , ∠K ≅ ∠Q,

∠L ≅ ∠R, and ∠M ≅ ∠S; So, △KLM ≅ △QRS because all six pairs of corresponding parts are congruent.

― AB ≅ ― DE ,

― BC ≅ ― EF , ― AC ≅ ― DF , ∠A ≅ ∠D,

∠B ≅ ∠E, and ∠C ≅ ∠F; So, △ABC ≅ △DEF because all six pairs of corresponding parts are congruent.

∠R ≅ ∠U, but m∠S ≠ m∠V and m∠T ≠ m∠W, so ∠S is not congruent to ∠V and ∠T is not congruent to ∠W. Not congruent; there aren’t six pairs of congruent corresponding parts.

― KL ≅ ― NP and

― JL ≅ ― MP . However, since JK ≠ MN,

― JK is not congruent to ― MN .

Not congruent; there aren’t six pairs of congruent corresponding parts.

― RS corresponds to

― MN .

RS = 1.8 cm.

3z - 3 = 1.8, so 3z = 4.8, and z = 1.6

∠H corresponds to ∠C. 50° + 69° + m∠C = 180° m∠C = 61° m∠H = 61° 2x + 5 = 61, so 2x = 56, and x = 28


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GE_MNLESE385795_U2M05L1.indd 225 2/26/16 2:33 AMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1 1 Recall Information MP.6 Precision

2–5 1 Recall Information MP.2 Reasoning

6–9 1 Recall Information MP.4 Modeling

10–21 2 Skills/Concept MP.2 Reasoning

22 3 Strategic Thinking MP.3 Logic

23 3 Strategic Thinkings MP.3 Logic

24 3 Strategic Thinking MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreTransforming Triangles with Congruent Corresponding Parts

Exercise 1

Example 1Deciding if Triangles are Congruent

Exercises 2–5

Example 2Applying the Properties of Congruent Triangles

Exercises 6–9

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Suggest that students list the six congruencies that relate the parts of the triangles and mark the triangles to show them. Once they have clearly represented the corresponding parts, they can more easily answer the questions.

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8. 9.

Determine whether the given triangles are congruent. Explain.

10.

11.

E

G

F

K

J

H

(7y - 10) mm31 mm

25 mm

24 mm

24 mm

31 mm

22°

22°

36°

36°

(4w + 14)°

XY

ZQ

R

P

E D

F

2

-4

4

20 4

x-4

A B

C

y

63.4°

26.6°

Q

R

P

2

4

4

x-2

-4

-4

K

L

M

y

0

_ EF corresponds to

_ HJ .

EF = 25 mm

7y - 10 = 25, so 7y = 35, and y = 5

∠Q corresponds to ∠Y.

22° + m∠Y + 36° = 180°

m∠Y = 122°

m∠Q = 122°

4w + 14 = 122 , so 4w = 108, and w = 27

Since _ AB is horizontal and

_ BC is vertical, ∠B is a right

angle, so m∠B = 90°.

Since DE is horizontal and EF is vertical, ∠E is a right

angle, so m∠E = 90°.

By the Triangle Sum Theorem, 63.4° + 90° + m∠C = 180°,

so m∠C = 26.6°.

Also, m∠D + 90° + 26.6° + 180°, so m∠D = 63.4°.

KL = √ ―――――――― (2 - (-2) ) 2 + (2 - 4) 2 = √ ― 20

LM = √ ――――――― (-4 - 2) 2 + (0 - 2) 2 = √ ― 40

KM = ―――――――――― (-4 - (-2) )

2 + (0 - 4) 2 = √ ― 20

PQ = √ ――――――――― (3 - (-1) ) 2 + (0 - (-2) )

2 = √ ― 20

PR = √ ―――――――――― (3 - (-1) ) 2 + (-4 - (-2) )

2 = √ ― 20

Therefore, ∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F.

_ AB is horizontal and its length is 3 units.

_ BC is vertical and its length is 6 units.

_ DE is horizontal and its length is 3 units.

_ EF is vertical and its length is 6 units.

By the Distance Formula, AC = √ ――――――――― (-1- (-4) ) 2 + (-3 - 3) 2 = √ ― 45 .

Also, DF = √ ――――――― (1 - 4) 2 + (-4 - 2) 2 = √ ― 45 .

Therefore, _ AB ≅

_ DE , _ BC ≅

_ EF , and

_ AC ≅

_ DF , so the triangles have six pairs of congruent

corresponding parts, and are congruent by the converse of CPCTC.

_ QR is vertical and its length is 4 units.

Therefore, _ KL ≅

_ PQ and KM ≅

_ PR , but LM ≠ QR, so LM is not congruent to

_ QR .

So, the triangles are not congruent.





AVOID COMMON ERRORSStudents may write congruence statements incorrectly. Make sure they understand that the order of the vertices in a congruence statement is not random. Students should know that they can identify corresponding angles by choosing pairs of letters in corresponding positions in a congruence statement.



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lva/

Cor

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12.

13.

14. △FGH represents an artist’s initial work on a design for a new postage stamp. What must be the values of x, y, and z in order for the artist’s stamp to be congruent to △ABC?

W

V

X

4

x-2-4

-4

S

UT

y

18.4°

74.1°

0 2 4

A

B

C

4

2 4

x-4

-4

FD

E

y

33.7°

56.3°

0

25 mm

25 mm

(5y) mm(3z + 14) mm

(2x + 5) mm

35 mm

F G

H A

C

B

Since ST is vertical and _ TU is horizontal, ∠T is a right angle,

so m∠T = 90°.

Since ― VW is vertical and

― WX is horizontal, ∠W is a right angle, so m∠W = 90°.

By the Triangle Sum Theorem, 74.1° + 90° + m∠U = 180°, so m∠U = 15.9°.

Also, m∠V + 90° + 18.4° = 180°, so m∠V = 71.6°.

Therefore, ∠T ≅ ∠W, but ∠S is not congruent to ∠V or ∠X.

So, the triangles are not congruent.

∠B is a right angle, so m∠B = 90°.

∠E is a right angle, so m∠E = 90°.

By the Triangle Sum Theorem,

33.7° + 90° + m∠C = 180°, so m∠C = 56.3°.

Also, m∠D + 90° + 56.3° = 180°, so m∠D = 33.7°.

Therefore, ∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F.

AB = √ ――――――――― (0 - (-3) ) 2 + (-2 - 1) 2 = √ ― 18 DE = √ ――――――― (0 - 3) 2 + (0 - 3) 2 = √ ― 18

BC = √ ――――――――― (-2 - 0) 2 + (-4 - (-2) ) 2 = √ ― 8 EF = √ ――――――― (-2 - 0) 2 + (2 - 0) 2 = √ ― 8

AC = √ ――――――――― (-2 - (-3) ) 2 + (-4 - 1) 2 = √ ― 26 DF = √ ――――――― (-2 - 3) 2 + (2 - 3) 2 = √ ― 26

Therefore _ AB ≅

_ DE , _ BC ≅

_ EF , and

_ AC ≅

_ DF , so the triangles have six pairs of congruent

corresponding parts, and are congruent by the converse of CPCTC.

∠F corresponds to ∠A, ∠G corresponds to ∠B, and ∠H corresponds to ∠C.

― FG corresponds to AB .

― GH corresponds to ― BC .

― FH corresponds to ― AC .

FG = 25 mm GH = 25 mm FH = 35 mm

2x + 5 = 25 5y = 25 3z + 14 = 35

x = 10 y = 5 z = 7


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


VISUAL CUESHave each student design a poster to illustrate the concept of congruent triangles. The illustrations should show both an example and a nonexample.

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15. Multi-Step Find the values of the variables that result in congruent triangles.

Determine whether each statement is always, sometimes, or never true. Explain your reasoning.

16. If △ABC has angles that measure 10° and 40°, and △DEF has angles that measure 40° and 120°, then △ABC ≅ △DEF.

17. Two triangles with different perimeters are congruent.

18. If △JKL ≅ △MNP, then m∠L = m∠N. 19. Two triangles that each contain a right angle are congruent.

20. Tenaya designed the earrings shown. She wants to be sure they are congruent. She knows that the three pairs of corresponding angles are congruent. What additional measurements should she make? Explain.

84° 84°

L

M

Q

R

(3x + 24)º

(7x - 4)º

(2x + y)º

P

N

INT_MCAESE216709_12AKenneth Batelman

37 mm35 mm

12 mm

A D

B EC F

∠L corresponds to ∠P.

m ∠L = m ∠P

3x + 24 = 7x - 4

3x + 28 = 7x

7= x

m∠L = (3x + 24)° = (3 ∙ 7 + 24)° = 45°

Never true; by the Triangle Sum Theorem, the third angle of △ABC measures 130° and the third angle of △DEF measures 20°. Since corresponding angles are not congruent, the triangles cannot be congruent.

45° + 84° + m∠N = 180°

m∠N = 51°

2x + y = 51

2 (7) + y = 51, So y = 37.

Never true; if the corresponding sides of two triangles are congruent, then the perimeters are equal. Therefore, if the perimeters are not equal, at least one pair of corresponding sides is not congruent, so the triangles are not congruent.

Sometimes true; ∠L corresponds to ∠P , so by CPCTC, ∠L ≅ ∠P. If ∠P ≅ ∠N, then ∠L ≅ ∠N by transitivity, and m∠L = m∠N.

Sometimes true; the right angles in the triangles are congruent corresponding angles. Other corresponding angles may or may not be congruent, and corresponding sides may or may not be congruent.

Corresponding sides must also be congruent. So, AB must equal DE, BC must equal EF, and AC must equal DF. She should check that AC = 35 mm, BC = 12 mm, and DE = 37 mm.







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21. Determine whether △JKL and △PQR are congruent or not congruent based on the given information. Select the correct answer for each lettered part.

a. m∠J = m∠K = m∠L = 60°, m∠P = m∠Q = m∠R = 60°, JK = KL = JL = 1.2 cm, PQ = QR = PR = 1.5 cm Congruent Not congruent

b. m∠J = 48°, m∠K = 93°, m∠P = 48°, m∠R = 39°, _ JK ≅ _ PQ , _ KL ≅ _ QR , _ JL ≅

_ PR Congruent Not congruent

c. ∠J ≅ ∠P, ∠K ≅ ∠Q, ∠L ≅ ∠R, JK = PQ = 22 in., KL = QR = 34 in., JL = PR = 28 in. Congruent Not congruent

d. m∠J = 51°, m∠K = 77°, m∠P = 51°, m∠R = 53° Congruent Not congruent

e. m∠J = 45°, m∠K = 80°, m∠Q = 80°, m∠R = 55°, JK = PQ = 1.5 mm, KL = QR = 1.3 mm, JL = PR = 1.8 mm Congruent Not congruent

H.O.T. Focus on Higher Order Thinking

22. Counterexamples Isaiah says it is not necessary to check all six pairs of congruent corresponding parts to decide whether two triangles are congruent. He says that it is enough to check that the corresponding angles are congruent. Sketch a counterexample. Explain your counterexample.

23. Critique Reasoning Kelly was asked to determine whether △KLN is congruent to △MNL. She noted that

_ KL ≅ _ MN ,

_ KN ≅ _ ML , and that the three pairs of corresponding angles

are congruent. She said that this is only five pairs of congruent corresponding parts, so it is not possible to conclude that △KLN is congruent to △MNL. Do you agree? Explain.

24. Analyze Relationships David uses software to draw two triangles. He finds that he can use a rotation and a reflection to map one triangle onto the other, and he finds that the image of vertex D is vertex L, the image of vertex V is vertex C, and the image of vertex W is vertex Y. In how many different ways can David write a congruence statement for the triangles? Explain.

44 in. 44 in.

72 in.

72 in.

K

N M

L

a. Not congruent; corresponding sides are not congruent.

b. Congruent; by the Triangle Sum Theorem, m∠L = 39°and m∠Q = 93°.

c. Congruent; there are six pairs of congruent corresponding parts.

d. Not congruent; by the Triangle Sum Theorem, m∠L = 52° and m∠Q = 76°.

e. Congruent; by the Triangle Sum Theorem, m∠L = 55° and m∠P = 45°.

DA

CBFE 60°

60°60°

60°

60° 60°

△ABC and △DEF have three pairs of congruent corresponding angles, but the triangles are not congruent because corresponding sides are not the same length.

Possible sketch:

No; _ LN ≅

― LN by the Reflexive Property of Congruence, so there are actually six pairs of congruent corresponding parts. Therefore, the triangles are congruent.

12 ways; △DVW ≅ △LCY, △DWV ≅ △LYC, △VDW ≅ △CLY, △VWD ≅ △CYL, △WDV ≅ △YLC, or △WVD ≅ △YCL, and 6 more with the order of the triangles reversed.




JOURNALHave students write a journal entry in which they explain in their own words what CPCTC means. Encourage them to include one or more labeled figures as part of the journal entry.

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For Kenny’s science project, he is studying whether honeybees favor one color of eight-petal flowers over other colors. For his display, he is making eight-petal flowers from paper in various colors. For each flower, he’ll cut out eight triangles like the one in the figure.

a. Find x, the measure in degrees of the top angle of each triangle. Explain how you found x.

b. Find y, the measure in degrees of the two base angles of each triangle. Explain how you found y.

c. Explain how Kenny could confirm that one of his triangles is congruent to the other seven.

Lesson Performance Task

y

x

y

a. To find x, use the fact that the sum of the measures of the eight angles at the center of the flower is 360°.

x = 360° ÷ 8 = 45°.

b. To find y, use the fact that the sum of the measures of the interior angles of a triangle is 180°.

y = 180° - 45° __ 2

= 135° _ 2

= 67.5°

c. Kenny could confirm that one of his triangles is congruent to the other seven by physically matching each of the seven to the first one to show that the three pairs of angles and the three pairs of sides are congruent. Or he could reason that one of the triangles can be mapped to each of the seven others through a series of 45° rotations.




EXTENSION ACTIVITY

Extend the Lesson Performance Task by completing the table shown for regular polygons with the given numbers of sides. Describe any patterns you observe. Possible patterns: As the number of sides

increases, the value of y increases, the value of x

decreases, and the ratio x __ y decreases; the sum

x + y + y always equals 180°.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Have students describe, in as much detail as possible, the numerical and geometric features of the 8-sided polygon in the Lesson Performance Task diagram. Sample answer: The figure is a regular

octagon. Drawing the four diagonals divides it into

8 isosceles triangles, each with base angles

measuring 67.5° and apex angle measuring 45°. The

8 sides of the octagon are congruent.

INTEGRATE MATHEMATICAL PRACTICES Focus on ReasoningMP.2 A regular polygon is divided into congruent isosceles triangles. One of the base angles of each isosceles triangle measures 84°. How many sides does the polygon have? Explain how you found the answer. Sample answer: Since each triangle is

isosceles, both base angles measure 84°. That leaves

180° – (84° + 84°) = 180° – 168° = 12° for the

remaining angle, the angle at the center of the

polygon. The sum of the angles at the center of the

polygon is 360°, so the polygon must have

360° ÷ 12° = 30 sides.

Number of Sides x y y

9 40 70 70

10 36 72 72

12 30 75 75

15 24 78 78

18 20 80 80

20 18 81 81

Scoring Rubric2 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.



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