CorrectionKey=NL-A;CA-A 6 . 2 DO NOT EDIT--Changes must be ...

Name Class Date

© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Explore Exploring Angle-Angle-Side Congruence

If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, are the triangles congruent?

In this activity you’ll be copying a side and two angles from a triangle.

A Use a compass and straightedge to copy segment AC. Label it as segment EF.

B Copy ∠A using _ EF as a side of the angle.

C On a separate transparent sheet or a sheet of tracing paper, copy ∠B. Label its vertex G. Make the rays defining ∠G longer than their corresponding sides on △ABC.

D Now overlay the ray from ∠E with the ray from ∠G to form a triangle. Make sure that side

_ EF

maintains the length you defined for it.

Resource Locker

E F

A C

B

E F

A C

B

A C

B

E F

G

A C

B

E F

G

A C

B

E F

G

A C

B

Module 6 283 Lesson 2

6.2 AAS Triangle CongruenceEssential Question: What does the AAS Triangle Congruence Theorem tell you about two

triangles?

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

GE_MNLESE385795_U2M06L2.indd 283 02/04/14 1:04 AM

Common Core Math StandardsThe student is expected to:

G-SRT.B.5

Use congruence ... criteria for triangles to solve problems and to prove relationships in geometric figures.

Mathematical Practices

MP.3 Logic

Language ObjectiveExplain in your own words the difference between the AAS and ASA congruence theorems.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 245254

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

AAS Triangle Congruence

ENGAGE Essential Question: What does the AAS Triangle Congruence Theorem tell you about two triangles?If two angles and a non-included side of one

triangle are congruent to the corresponding angles

and side of another triangle, the triangles are

congruent.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the figure, making sure students understand why a triangle with the given side lengths cannot be drawn. Then preview the Lesson Performance Task.

283

HARDCOVER

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

Name

Class Date

© H

ough

ton

Mif

flin

Har

cour

t Pub

lishi

ng C

omp

any

Explore Exploring Angle-Angle-Side

Congruence

If two angles and a non-included side of one triangle are congruent to the corresponding angles

and side of another triangle, are the triangles congruent?

In this activity you’ll be copying a side and two angles from a triangle.

Use a compass and straightedge to copy

segment AC. Label it as segment EF.

Copy ∠A using _ EF as a side of the angle.

On a separate transparent sheet or a sheet of

tracing paper, copy ∠B. Label its vertex G.

Make the rays defining ∠G longer than their

corresponding sides on △ABC.

Now overlay the ray from ∠E with the ray from

∠G to form a triangle. Make sure that side _ EF

maintains the length you defined for it.

Resource

Locker

G-SRT.B.5 Use congruence ... criteria for triangles to solve problems and to prove relationships in

geometric figures.COMMONCORE

E

F

A

C

B

E

F

A

C

B A

C

B

E

F

GA

C

B

EF

GA

C

B

EF

GA

C

B

Module 6

283

Lesson 2

6 . 2 AAS Triangle Congruence

Essential Question: What does the AAS Triangle Congruence Theorem tell you about two

triangles?

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

CorrectionKey=NL-A;CA-A

GE_MNLESE385795_U2M06L2.indd 283

02/04/14 1:05 AM

283 Lesson 6 . 2

L E S S O N 6 . 2

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

© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

E How many triangles can you construct?

F Copy all of △EFG to the transparency. Then overlay it on △ABC. Are the triangles congruent? How do you know?

Reflect

1. Suppose you had started this activity by copying segment BC and then angles A and C. Would your results have been the same? Why or why not?

2. Compare your results to those of your classmates. Does this procedure work with any triangle?

Explain 1 Justifying Angle-Angle-Side CongruenceThe following theorem summarizes the previous activity.

Angle-Angle-Side (AAS) Congruence Theorem

If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent.

Prove the AAS Congruence Theorem.

Given: ∠A ≅ ∠D, ∠C ≅ ∠F, _ BC ≅

_ EF

Prove: △ABC ≅ △DEF

Statements Reasons

1. ∠A ≅ ∠D, ∠C ≅ ∠ , _

BC ≅ _

EF 1. Given

2. m∠A + m∠B + m∠C = 180° 2.

3. m∠B = 180° - m∠A - m∠ 3. Subtraction Property of Equality

4. m∠ + m∠E + m∠F = 180° 4. Triangle Sum Theorem

5. m∠E = 180° - m∠D - m∠ 5. Subtraction Property of Equality

6. m∠A = m∠D, m∠C = m∠F 6.

7. m∠E = 180° - m∠A - m∠C 7.

8. m∠ = m∠B 8. Transitive Property of Equality

9. ∠E ≅ ∠B 9.

10. △ABC ≅ △DEF 10. Triangle Congruence Theorem

A B

C

D E

F

Only one triangle is possible.

Yes; possible answer: All sides are

congruent, so they are congruent by SSS.

Yes; no specific side lengths and angle measures were involved.

Yes. All triangles with congruent angles and a congruent non-included side are congruent.

F

C

D

F

E

Definition of congruent angles

Substitution

Definition of congruent angles

ASA

Triangle Sum Theorem


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

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

GE_MNLESE385795_U2M06L2.indd 284 2/26/16 12:23 AM

Integrate Mathematical PracticesThis lesson provides an opportunity to address Mathematical Practice MP.3, which calls for students to “construct viable arguments and critique the reasoning of others.” Students learn to justify the Angle-Angle-Side Congruence Theorem by using a paragraph proof. Mathematical proofs must use precise language to ensure their validity. As students continue to explore congruent triangles, ask them to justify their conclusions and communicate them with the class. Promoting this type of dialogue in the classroom is an essential aspect of the standard.

EXPLORE Exploring Angle-Angle-Side Congruence

INTEGRATE TECHNOLOGYStudents can use geometry software to explore how much they know about a triangle when given two angle measures and the length of a nonincluded side.

QUESTIONING STRATEGIESIf you know the measures of two angles in a triangle, and the length of a side that is not

included, does this describe a unique triangle? What is the relationship between the information necessary to describe a unique triangle and the information necessary to prove two triangles congruent? Yes; the

same sides and angles that must be known to be

congruent to prove two triangles congruent must

be given to describe a unique triangle (two angles

and the included side, two sides and the included

angle, etc.).

EXPLAIN 1 Justifying Angle-Angle-Side Congruence

QUESTIONING STRATEGIESIf you know the measures of two angles of a triangle, what theorem gives you a way to

determine the measure of the third angle? Third

Angles Theorem

AVOID COMMON ERRORSAfter finding the measure of the third angle, some students may want to say the triangles are congruent by AAA. Remind them that there is no AAA congruence theorem because having three pairs of congruent angles does not ensure two triangles have the same size.

PROFESSIONAL DEVELOPMENT

AAS Triangle Congruence 284


© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Reflect

3. Discussion The Third Angles Theorem says “If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.” How could using this theorem simplify the proof of the AAS Congruence Theorem?

4. Could the AAS Congruence Theorem be used in the proof? Explain.

Explain 2 Using Angle-Angle-Side Congruence

Example 2 Use the AAS Theorem to prove the given triangles are congruent.

Given: _ AC ≅

_ EC and m‖n

Prove: △ABC ≅ △EDC

Given: _ CB ‖ _ ED , _ AB ‖ _ CD , and

_ CB ≅ _ ED .

Prove: △ABC ≅ △CDE

Given

Alt. Int � Thm.

Given

Alt. Int � Thm.

AC ≅ EC

∠E ≅ ∠A

AAS Cong. Thm.

△ABC ≅ △EDC

∠B ≅ ∠D

m ǁ n

A

B

D

C

E

Given

Corr. � Thm.

Given Corr. � Thm.

CB ≅ ED

∠ACB ≅ ∠CED

AAS Cong. Thm.

△ABC ≅ △CDE

∠CAB ≅ ∠ECD

Given

CB ǁ ED

AB ǁ CD

A

C

B

m

n

D E

Steps 2–8 could be simplified to one step, ∠B ≅ ∠E, making the whole

proof only three steps.

No, that is the theorem being proved, so it cannot be used until it has

been proven.


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

GE_MNLESE385795_U2M06L2 285 5/22/14 9:04 PM

COLLABORATIVE LEARNING

Small Group ActivityStudents may benefit from exploring conditions that do not guarantee congruent triangles. Have students work in small groups and use a protractor to draw triangles with angles that measure 35°, 50°, and 95°. Then then compare their triangles within each group. Students will see that all the triangles have the same shape, but not necessarily the same size. Since the triangles are not all congruent, the activity shows that there is no AAA Congruence Criteria.

EXPLAIN 2 Using Angle-Angle-Side Congruence

QUESTIONING STRATEGIESHow does the AAS Congruence Theorem differ from the ASA Congruence

Theorem? While both the ASA and AAS Congruence

Theorems require you to know that two pairs of

corresponding angles are congruent between two

triangles, the ASA Congruence Theorem requires

you to also know that a pair of corresponding

included sides are congruent. The AAS Congruence

Theorem requires you to know also that a pair of

corresponding non-included sides are congruent.

If AAS is used as the method of proof, can the triangles also be proved congruent using

ASA? Yes; you can use the Third Angles Theorem to

show the third pair of angles is congruent, and then

you can use ASA.

285 Lesson 6 . 2


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn

5. Given: ∠ABC ≅ ∠DEF, _ BC ∥ _ EF , _ AC ≅

_ DF . Use the AAS Theorem to prove the

triangles are congruent. Write a paragraph proof.

Explain 3 Applying Angle-Angle-Side Congruence

Example 3 The triangular regions represent plots of land. Use the AAS Theorem to explain why the same amount of fencing will surround either plot.

Given: ∠A ≅ ∠D

It is given that ∠A ≅ ∠D. Also, ∠B ≅ ∠E because both are right angles. Compare AC and DF using the Distance Formula.

AC = √ ―――――――― ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

= √ ――――――――― (-1- (-4) ) 2 + (4 - 0) 2

= √ ――― 3 2 + 4 2

= √ ― 25

= 5

DF = √ ―――――――― ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

= √ ――――――― (4 - 0) 2 + (1 - 4) 2

= ―――― 4 2 + (-3) 2

= √ ― 25

= 5

Because two pairs of angles and a pair of non-included sides are congruent, △ABC ≅ △DEF by AAS. Therefore the triangles have the same perimeter and the same amount of fencing is needed.

A

B

C D F

E

y

x

2 4-2-2

-4

-4

4

2

0A

B C D E

F

Because ̄ BC is parallel to ̄ EF , this means that ∠ACB is congruent to ∠DFE, using the Corresponding Angles Theorem. Since ∠ABC is congruent to ∠DEF and ̄ AC is congruent to ̄ DF , then one pair of corresponding angles and two pairs of non-included corresponding sides are congruent. This means that △ABC is congruent to △DEF using the AAS Triangle Congruence Theorem.


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

GE_MNLESE385795_U2M06L2 286 6/16/15 10:23 AM

DIFFERENTIATE INSTRUCTION

Auditory CuesAs students work through the lesson, have them identify orally whether the triangles are congruent by ASA, AAS, SAS, or SSS. Then have them work with a partner to identify the congruent parts and to determine if the congruent pairs of angles or sides are corresponding parts.

EXPLAIN 3 Applying Angle-Angle-Side Congruence

AVOID COMMON ERRORSThe real-world situation may distract some students. Make sure they are certain what they are solving for before they begin.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

B Given: ∠P ≅ ∠Z, ∠Q ≅ ∠X

It is given that ∠P ≅ ∠Z and ∠Q ≅ ∠X.

Compare YZ and using the distance formula.

YZ = √ ―――――――― ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

= ――――――――――――

( (-1) - ) 2

+ ( (-2) - ) 2

= ――――――

( ) 2

+ ( ) 2

=

――――― +

= √ ――

= √ ―――――――― ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

=

――――――――― ( - 0)

2

+ ( - 0) 2

= ―――――――

( ) 2

+ ( ) 2

=

――――― +

= √ ――

Because two pairs of angles and a pair of non-included sides are congruent,

△XYZ ≅ △ by AAS. Therefore the triangles have the same perimeter and the same amount of fencing is needed.

Reflect

6. Explain how you could have avoided using the distance formula in Example 2B.

X

Y

Z

P

QO

2 4-4

4

2

-4

0 x

y

OP

OP

-5 -4

-4

QOP

-4

2

2

4 2

4

4

16

16

20

20

I could have used ̄ XY and ̄ QO as corresponding sides. By counting

gridlines, it is easy to see that XY = QO = 7. These are also not the

included sides between the angles, so AAS applies.


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

GE_MNLESE385795_U2M06L2 287 6/16/15 10:27 AM

LANGUAGE SUPPORT

Communicate MathDivide students into pairs. Give each pair pictures of congruent andnon-congruent triangles, highlighters, protractors, and rulers. Instruct them to prove which pairs are congruent by angle-angle-side. Have students highlight the angles and the side they used to show congruence, and write notes explaining why the triangles are congruent.

QUESTIONING STRATEGIESWhen is the distance formula useful for proving two triangles to be congruent? How

would you use it? It is useful when the triangles are

on a coordinate grid and their side lengths are not

given; you would use it to find the lengths of

corresponding sides and determine whether or not

the sides are congruent.

287 Lesson 6 . 2


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Your Turn

Refer to the diagram to answer the questions.

Given: ∠A ≅ ∠D and ∠B ≅ ∠E

7. Show that the two triangles are congruent using the AAS Theorem. Use the distance formula to compare BC and EF.

8. Show that the two triangles are congruent using the AAS Theorem. Use the distance formula to compare AC and DF.

A

E

F

D

B

C

y

2 4-4 -2

4

2

-4

-2

0x

It is given that ∠A ≅ ∠D and ∠B ≅ ∠E.

BC = ―――

(-5 - (-1) ) 2 + (5 - 2) 2 EF =

――― (1 - 5) 2 + (2 - (-1) )

2

= √ ――― 16 + 9 = √ ――― 16 + 9

= 5 = 5

Sides BC and EF are equal. Therefore, because two angles and a non-included side are congruent, △ABC ≅ △DEF by the AAS Theorem.

It is given that ∠A ≅ ∠D and ∠B ≅ ∠E.

AC = ―――

(-4 - (-1) ) 2 + (0 - 2) 2 DF =

――― (2 - 5) 2 + (-3 - (-1) )

2

= √ ―― 9 + 4 = √ ―― 9 + 4

= √ ― 13 = √ ― 13

Sides AC and DF are equal. Therefore, because two angles and a non-included side are congruent, △ABC ≅ △DEF by the AAS Theorem.




GE_MNLESE385795_U2M06L2.indd 288 3/20/14 4:20 PM

CONNECT VOCABULARY Students can create a chart, or add to a chart, proving triangle congruence, defining angle/angle/side, and showing a picture that demonstrates this condition.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Elaborate

9. Two isosceles triangles share a side. With which diagram can the AAS Theorem be used to show the triangles are congruent? Explain.

10. What must be true of the right triangles in the roof truss to use the AAS Congruence Theorem to prove the two triangles are congruent? Explain.

11. Essential Question Check-In You know that a pair of triangles has two pairs of congruent corresponding angles. What other information do you need to show that the triangles are congruent?

B

A C

D

The one on the right. The shared side is the non-included side.

Because ∠BDA and ∠BDC are right angles, ∠BDA ≅ ∠BDC. Since _ BD is a shared side for

both triangles, it must be the non-included side. So to use the AAS Theorem, ∠BAD must

be congruent to ∠BCD. This gives two angles and a non-included side.

You need a pair of congruent sides that are either both included or both

not included.




ELABORATE INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Students may benefit from seeing the proof of the AAS Congruence Criterion presented as a flowchart proof.

SUMMARIZE THE LESSONWhen do you use the AAS Triangle Congruence Theorem? What information do

you need in order to use this theorem? You use the

AAS Triangle Congruence Theorem to prove that

two triangles are congruent by using three pairs of

congruent corresponding parts. You need to know

that two pairs of corresponding angles and one pair

of corresponding non-included sides are congruent

in order to use it.

289 Lesson 6 . 2


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Evaluate: Homework and Practice

Decide whether you have enough information to determine that the triangles are congruent. If they are congruent, explain why.

1. 2.

3. 4.

5. 6.

Each diagram shows two triangles with two congruent angles or sides. Identify one additional pair of corresponding angles or sides such that, if the pair were congruent, the two triangles could be proved congruent by AAS.

7. 8.

9. 10.

11. 12.

• Online Homework• Hints and Help• Extra Practice

A

B

D

E

FC A

D

E

F

C

B

AD

EF

BC

A

BC

D

EF

AD E

FBC

A D

E

F

B C

A

B

CD

E

F

A

D

E

FCB

A

B CD

E F

A

B

C

D

E

F

A

B

C

D

EA

B

DC

Congruent, by AAS Congruence Congruent, by AAS Congruence

Congruent, by ASA Congruence Cannot be determined.

Cannot be determined. Congruent, AAS Congruence

∠A ≅ ∠E _ AB ≅

_ DE , or

_ BC ≅

_ EF

_ AB ≅

_ DE , or

_ BC ≅

_ DC

_ AB ≅

_ DE , _ AB ≅

_ DF , _ AC ≅

_ DE , or

_ AC ≅

_ DF ∠A ≅ ∠E

∠A ≅ ∠D


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

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

GE_MNLESE385795_U2M06L2.indd 290 3/20/14 4:20 PMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1–12 1 Recall of information MP.6 Precision

13–21 2 Skills/Concepts MP.4 Modeling

22 3 Strategic Thinking MP.2 Reasoning



EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreExploring Angle-Angle-Side Congruence

Exercises 1–12

Example 1Justifying Angle-Angle-Side Congruence

Exercises 13–14

Example 2Using Angle-Angle-Side Congruence

Exercises 13–14

Example 3Applying Angle-Angle-Side Congruence

Exercises 15–21

QUESTIONING STRATEGIESIf you know the measures of two angles of a triangle, how can you find the measure of the

third angle? Find the difference between 180° and

the sum of the two known angle measures.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

13. Complete the proof.

Given: ∠B ≅ ∠D, ‹ −

› AC bisects ∠BCD.

Prove: △ABC ≅ △ADC

14. Write a two-column proof or a paragraph proof.

Given: _ AB ∥ _ DE , _ CB ≅ _ CD .

Prove: △ABC ≅ △EDC

Each diagram shows △ABC and △DEF on the coordinate plane, with ∠A ≅ ∠E, and ∠C ≅ ∠F. Identify whether the two triangles are congruent. If they are not congruent, explain how you know. If they are congruent, find the length of each side of each triangle.

15. 16.

Statements Reasons

1. _

AC ≅ _

AC 1.

2. ‹ −

› AC bisects ∠BCD. 2. Given

3. 3. Definition of angle bisector

4. 4. Given

5. △ABC ≅ △ADC 5.

A

B

C

D

A

BC

D

E

A

B C

D E

F

2

0

4

2 4x

-2-4

y

AB

C

E

D

F1 x-2-4 4

y

Statements Reasons

1. _ AB ∥ _ DE , _ CB ≅

_ CD 1. Given

2. ∠A ≅ ∠E, ∠B ≅ ∠D 2. Alternate Interior Angles Theorem

3. △ABC ≅ △EDC 3. AAS Triangle Congruence Theorem

Reflexive Property of Congruence

AAS Triangle Congruence Theorem

∠ACB ≅ ∠ACD

∠B ≅ ∠D

Congruent; AB = DE = 4; BC = DF = 2; AC = EF = 4.47

Congruent. AB = ED = 3, AC = EF = 4, BC = DF = 5



GE_MNLESE385795_U2M06L2.indd 291 3/20/14 4:20 PMExercise Depth of Knowledge (D.O.K.) COMMONCORE Mathematical Practices

25 3 Strategic Thinking MP.3 Logic

26 3 Strategic Thinking MP.3 Logic

27 3 Strategic Thinking MP.4 Modeling

VISUAL CUESHave students each make a poster on which they describe and illustrate the AAS theorem. The illustrations should show both an example and a nonexample.

291 Lesson 6 . 2


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

17. 18.

19. 20.

21. Which theorem or postulate can be used to prove that the triangles are congruent? Select all that apply.

A. ASA B. SAS C. SSS D. AAS

H.O.T. Focus on Higher Order Thinking

22. Analyze Relationships △XYZ and △KLM have two congruent angles: ∠X ≅ ∠K and ∠Y ≅ ∠L. Can it be concluded that ∠Z ≅ ∠M? Can it be concluded that the triangles are congruent? Explain.

23. Communicate Mathematical Ideas △GHJ and △PQR have two congruent angles: ∠G ≅ ∠P and ∠H ≅ ∠Q. If

_ HJ is congruent to one of the sides of △PQR, are

the two triangles congruent? Explain.

A

B

C

D

E

F

4

0 4 8x

-4

y

A

B

C

DE

F

4

0 4 8x

y

A

C

B

D

E F4

0

8

4 8 12x

-8-12

-8

-4

y A

B

C

E

D

F

2

4

2 4 6 8x

-4

-2

y

Not congruent. Segments ̄ AB and ̄ DF are corresponding, but they have different lengths (AB = 4, DF = 5) .

Congruent; AB = DE = 2.8; BC = DF = 8; AC = EF = 6.3

Not congruent. Segments ̄ AC and ̄ EF are corresponding, but they have different lengths (AC = 9, EF = 10) .

Congruent. AB = DE = 10.29, AC = EF = 9.9, BC = DF = 2.8

It can be concluded that ∠Z ≅ ∠M, because the sum of the three angles of a triangle equals 180°. While the two triangles are similar, it cannot be concluded that the triangles are congruent without identifying at least one congruent side.

They are not necessarily congruent. Only if _ HJ is congruent to the

corresponding side, which is _ QR , will the triangles be congruent.


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


GE_MNLESE385795_U2M06L2 292 5/23/14 11:12 AM

AVOID COMMON ERRORSSome students may think that two pairs of congruent angles and any pair of non-included sides is enough to use AAS. They need to be sure the non-included sides are corresponding sides, otherwise they may be looking at triangles that are not congruent.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

24. Make a Conjecture Combine the theorems of ASA Congruence and AAS Congruence into a single statement that describes a condition for congruency between triangles.

25. Justify Reasoning Triangles ABC and DEF are constructed with the following angles: m∠A = 35°, m∠B = 45°, m∠D = 65°, m∠E = 45°. Also, AC = DF = 12 units. Are the two triangles congruent? Explain.

26. Justify Reasoning Triangles ABC and DEF are constructed with the following angles: m∠A = 65°, m∠B = 60°, m∠D = 65°, m∠F = 55°. Also, AB = DE = 7 units. Are the two triangles congruent? Explain.

27. Algebra A bicycle frame includes △VSU and △VTU, which lie in intersecting planes. From the given angle measures, can you conclude that △VSU ≅ △VTU? Explain.

m∠VUS = (7y - 2) ° m∠VUT = (5 1 _ 2 x - 1 _ 2 ) ° m∠USV = 5 2 _ 3 y° m∠UTV = (4x + 8 ) °

m∠SVU = (3y - 6) ° m∠TVU = 2x°

U

V

U

SS

TT

V

If two triangles have two congruent angles and one corresponding congruent side, then the two triangles are congruent.

The two triangles cannot be congruent. Because the sum of the three angles of a triangle equals 180°, it can be calculated that m∠C = 100° and that m∠F = 70°. The triangles do not have matching angles, so they are not congruent.

The two triangles are congruent by either AAS or ASA Congruence. The triangles each have angles of 65°, 60°, and 55°, and each has a side of length 7 that is opposite the angle of 55°.

Yes, the sum of the ∠ measures in each triangle must be 180°, which makes it possible to solve for x and y. The value of x is 15, and the value of y is 12. Each triangle has angles measuring 82°, 68°, and 30°.

_ VU ≅

_ VU by

the Reflexive Property of Congruence. So △VSU ≅ △VTU by ASA or AAS.



GE_MNLESE385795_U2M06L2 293 10/15/14 12:22 PM

JOURNALHave students write a journal entry in which they explain and illustrate the AAS Theorem.

293 Lesson 6 . 2


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Lesson Performance TaskA mapmaker has successfully mapped Carlisle Street and River Avenue, as shown in the diagram. The last step is to map Beacon Street correctly. To save time, the mapmaker intends to measure just one more angle or side of the triangle.

a. Which angle(s) or side(s) could the mapmaker measure to be sure that only one triangle is possible? For each angle or side that you name, justify your answer.

b. Suppose that instead of measuring the length of Carlisle Street, the mapmaker measured ∠A and ∠C along with ∠B. Would the measures of the three angles alone assure a unique triangle? Explain.

48°B

A

C

Beacon Street

River Avenue

Carlisle Street

0.4 mi

a. ∠C, by the AAS Triangle Congruence Theorem; ∠A, by the ASA Triangle Congruence Theorem;

_ BC , by the SAS Triangle Congruence Theorem

b. No; there is no AAA Triangle Congruence Theorem. There are many similar triangles with the same angle measures.




60 yd60 yd

30°

100 yd

EXTENSION ACTIVITY

A surveyor has a contract to lay out the boundaries of a triangular park. Here are the instructions given to the surveyor. (All of the streets are straight.)

• The vertex of one angle of the park will be the 30° angle where Charles Street and Barton Street intersect.

• One side of the park will start at the Charles-Barton vertex and extend 100 yards along Charles Street.

• The side of the park opposite the 30° angle will connect Barton Street and Charles Street and measure 60 yards.

What problem did the surveyor run into? Explain and draw a diagram.There are two possible triangles that meet the given conditions.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Draw attention to the figure in the Lesson Performance Task. State that m∠CAB = 85°. Ask students to imagine that the 48° angle where Carlisle Street and River Avenue intersect increases in size until Carlisle Street and Beacon Street no longer intersect. Ask: “What is the smallest measure of ∠B for which this would be possible? Explain your reasoning.” 133°; Sample answer: The smallest

measure of ∠B for which this would be possible

would be when the streets became parallel.

∠B and ∠ACB would be supplementary interior

angles on the same side of the River Avenue

transversal: m∠ACB = 180° ̵ (48° + 85°) = 47°. So, m∠B = 180° ̵ 47° = 133°.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Ask students to explain how they could use equilateral triangles to show that AAA Congruence does not hold. Sample answer: All equilateral

triangles from small ones on a company’s logo to

large “Yield” traffic signs share three pairs of

congruent (60°) angles. Yet the triangles are clearly

not congruent.

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.



CorrectionKey=NL-A;CA-A 6 . 2 DO NOT EDIT--Changes must be ...

Documents

Transcript of CorrectionKey=NL-A;CA-A 6 . 2 DO NOT EDIT--Changes must be ...