Proving Triangle Congruence by SAS

5.3 Proving Triangle Congruence by SAS

Essential Question What can you conclude about two triangles

when you know that two pairs of corresponding sides and the corresponding

included angles are congruent?

USING TOOLS STRATEGICALLY

To be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.

Drawing Triangles

Work with a partner. Use dynamic geometry software.

a. Construct circles with radii of 2 units and 3 units centered at the origin. Construct a 40? angle with its vertex at the origin. Label the vertex A.

b. Locate the point where one ray of the angle intersects the smaller circle and label this point B. Locate the point where the other ray of the angle intersects the larger circle and label this point C. Then draw ABC.

4

3

2

1

40?

0

A -4 -3 -2 -1

01

2

3

4

5

-1

-2

-3

4

c. Find BC, mB, and mC.

d. Repeat parts (a)?(c) several times, redrawing the angle in different positions. Keep track of your results by copying and completing the table below. What can you conclude?

3

2B

1

C

40?

0

A -4 -3 -2 -1

01

2

3

4

5

-1

-2

-3

A

B

1. (0, 0)

2. (0, 0)

3. (0, 0)

4. (0, 0)

5. (0, 0)

C AB AC BC mA mB mC

2

3

40?

2

3

40?

2

3

40?

2

3

40?

2

3

40?

Communicate Your Answer

2. What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent?

3. How would you prove your conclusion in Exploration 1(d)?

Section 5.3 Proving Triangle Congruence by SAS 245

5.3 Lesson

Core Vocabulary

Previous congruent figures rigid motion

STUDY TIP

The included angle of two sides of a triangle is the angle formed by the two sides.

What You Will Learn

Use the Side-Angle-Side (SAS) Congruence Theorem. Solve real-life problems.

Using the Side-Angle-Side Congruence Theorem

Theorem

Theorem 5.5 Side-Angle-Side (SAS) Congruence Theorem

If two sides and the included angle of one triangle are congruent to two sides and

the included angle of a second triangle, then the two triangles are congruent.

If A--B D--E, A D, and A--C D--F,

then ABC DEF.

B

E

F

Proof p. 246

C

AD

Side-Angle-Side (SAS) Congruence Theorem

Given A--B D--E, A D, A--C D--F

Prove ABC DEF

B C

AD

E F

First, translate ABC so that point A maps to point D, as shown below.

E

E

B

B

F

D

F

C

AD

C

This translation maps ABC to DBC. Next, rotate DBC counterclockwise

through CDF so that the image of DC coincides with DF, as shown below.

E

E

B

D

F

F

C

D

B

Because D--C D--F, the rotation maps point C to point F. So, this rotation maps

DBC to DBF. Now, reflect DBF in the line through points D and F, as

shown below.

E

E

F

D B

F D

Because points D and F lie on DF, this reflection maps them onto themselves. Because

atoreDflEe.cBtioencapurseesDe--rBve sanD--gEle,

measure and BDF EDF, the reflection maps DB

the reflection maps point B to point E. So, this reflection

maps DBF to DEF.

Because you can map ABC to DEF using a composition of rigid motions, ABC DEF.

246 Chapter 5 Congruent Triangles

STUDY TIP

Make your proof easier to read by identifying the steps where you show congruent sides (S) and angles (A).

Using the SAS Congruence Theorem

Write a proof.

Given B--C D--A, B--C A--D

Prove ABC CDA

SOLUTION

B A

C D

STATEMENTS

S 1. B--C D--A 2. B--C A--D

REASONS 1. Given 2. Given

A 3. BCA DAC

S 4. A--C C--A

3. Alternate Interior Angles Theorem (Thm. 3.2) 4. Reflexive Property of Congruence (Thm. 2.1)

5. ABC CDA

5. SAS Congruence Theorem

Using SAS and Properties of Shapes

In the diagram, Q--S and R--P pass through the center M of the circle. What can you

conclude about MRS and MPQ?

S

R

M

P

Q

SOLUTION

Because distance

they are from the

vertical center,

saonMg--lePs,,M-- QP,MM--QR, andRM--MSS.aArellalplocionntsgoruneantc.ircle

are

the

same

So, MRS and MPQ are congruent by the SAS Congruence Theorem.

Monitoring Progress

Help in English and Spanish at

IaannndgthlSe--esV.dRia,gVS--r,UaTm., ,aAnBd CUDariseathsequmairdepwoiinthtsfoofutrhceosnigdreuseonftAsBidCesDa. nAdlsfoo,uR--rTrighS--tU

B

S

C

R

V

T

A

U

D

1. Prove that SVR UVR. 2. Prove that BSR DUT.

Section 5.3 Proving Triangle Congruence by SAS 247

Step 1

Copying a Triangle Using SAS

Construct a triangle that is congruent to ABC using the

C

SAS Congruence Theorem. Use a compass and straightedge.

SOLUTION Step 2

Step 3 F

A

B

Step 4 F

D

E

CCisoocnnossnttrgrurucuctetDn--atEtsoisdoA--eBth.at it

D

E

Construct an angle Construct D with vertex

D and side DE so that it is

congruent to A.

D

E

CCit ooisnnscsttorrunucgctrtuD--eaFnstisdtooetA--hCat.

D

E

Draw a triangle Draw DEF. By the SAS Congruence Theorem, ABC DEF.

Solving Real-Life Problems

Solving a Real-Life Problem

You are making a canvas sign to hang

on the triangular portion of the barn wall

R

shown in the picture. You think you can

use two identical triangular sheets of

Pc--aQnvasP--. YSo. uUksenothwe

that R--P Q--S and

SAS Congruence

Q P

S

Theorem to show that PQR PSR.

SOLUTION

RY--PouarR--ePg.ivBeyn

that P--Q P--S. By the Reflexive Property of Congruence (Theorem 2.1),

the definition of perpendicular lines, both RPQ and RPS are right

angles, so they are congruent. So, two pairs of sides and their included angles are

congruent.

PQR and PSR are congruent by the SAS Congruence Theorem.

Monitoring Progress

Help in English and Spanish at

3.

cYoonugarureendtetsoigniDngRGth.eYwoiunddeoswigsnhothwenwiinntdhoewpshoottoh.aYt Do--uAwanD--t Gto

make and

DRA

ADR GDR. Use the SAS Congruence Theorem to prove DRA DRG.

D

248 Chapter 5 Congruent Triangles

A

R

G

5.3 Exercises

Dynamic Solutions available at

Vocabulary and Core Concept Check

1. WRITING What is an included angle?

2. COMPLETE THE SENTENCE If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then ___________.

Monitoring Progress and Modeling with Mathematics

In Exercises 3?8, name the included angle between the pair of sides given.

J

L

In Exercises 15?18, write a proof. (See Example 1.)

15. Given P--Q bisects SPT, S--P T--P

Prove SPQ TPQ

P

K

3. J--K and K--L 5. L--P and L--K 7. K--L and J--L

P

4. P--K and L--K 6. J--L and J--K 8. K--P and P--L

In Exercises 9?14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem (Theorem 5.5). Explain.

9. ABD, CDB

10. LMN, NQP

A B

D

L

C

M

Q N

P

11. YXZ, WXZ

12. QRV, TSU

Z

R

S

W

X

Y

Q

V

U

T

13. EFH, GHF

14. KLM, MNK

F

E

K

L

G

H

N

M

S

T

Q

16. Given A--B C--D, A--B C--D

Prove ABC CDA

A

D

1

2

B

C

17. Given C is the midpoint of A--E and B--D.

Prove ABC EDC

D

A

C

E

B

18. Given P--T R--T, Q--T S--T

Prove PQT RST

P

Q

T

S

R

Section 5.3 Proving Triangle Congruence by SAS 249

In Exercises 19?22, use the given information to name two triangles that are congruent. Explain your reasoning. (See Example 2.)

19. SRT URT, and 20. ABCD is a square with

R is the center of

four congruent sides and

the circle.

four congruent angles.

S T

B

C

R

U

21. RSTUV is a regular pentagon.

T

S

U

R

V

A

D

22. M--K M--N, K--L N--L,

and M and L are centers of circles.

K

10 m

M

L

10 m

N

CONSTRUCTION In Exercises 23 and 24, construct a triangle that is congruent to ABC using the SAS Congruence Theorem (Theorem 5.5).

23. B

24. B

A

C

A

C

25. ERROR ANALYSIS Describe and correct the error in finding the value of x.

Y 4x + 6

5x - 1 W

X 5x - 5 Z 3x + 9

4x + 6 = 3x + 9 x + 6 = 9 x = 3

26. HOW DO YOU SEE IT?

B

What additional information

do you need to prove that

ABC DBC?

A

C

D

27. PROOF The Navajo rug is made of isosceles triangles. You know B D. Use the SAS Congruence Theorem (Theorem 5.5) to show that ABC CDE. (See Example 3.)

B

D

A

C

E

28. THOUGHT PROVOKING There are six possible subsets of three sides or angles of a triangle: SSS, SAS, SSA, AAA, ASA, and AAS. Which of these correspond to congruence theorems? For those that do not, give a counterexample.

29. MATHEMATICAL CONNECTIONS Prove that

ABC DEC. Then find the values of x and y.

A 4y - 6 2x + 6 D C

B 3y + 1 4x E

30. MAKING AN ARGUMENT Your friend claims it is

possible to construct a triangle

ccoonngstrruuecntitntgoA--BAaBnCd Ab--yCf,iarsntd

C

then copying C. Is your

friend correct? Explain

A

B

your reasoning.

31. PROVING A THEOREM Prove the Reflections in Intersecting Lines Theorem (Theorem 4.3).

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Classify the triangle by its sides and by measuring its angles. (Section 5.1)

32.

33.

34.

35.

250 Chapter 5 Congruent Triangles

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download