8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 1/21

4.4 Proving Triangles are

Congruent: ASA and

AASGeometry

Mr. Moen

2011

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 2/21

Objectives:

1. Prove that triangles are congruent

using the ASA Congruence Postulate

and the AAS Congruence Theorem

2. Use congruence postulates and

theorems in real-life problems.

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 3/21

Postulate 21: Angle-Side-Angle

(ASA) Congruence Postulate

� If two angles and the

included side of one

triangle arecongruent to two

angles and the

included side of a

second triangle, thenthe triangles are

congruent.

B

C

A

F

D

E

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 4/21

Theorem 4.5: Angle-Angle-Side

(AAS) Congruence Theorem

� If two angles and anon-included side of 

one triangle arecongruent to twoangles and thecorresponding non-included side of asecond triangle, thenthe triangles arecongruent.

B

C

A

F

D

E

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 5/21

Theorem 4.5: Angle-Angle-Side

(AAS) Congruence Theorem

Given: �A $ �D, �C

$ �F, BC $ EF

Prove: ¨ABC $ ¨DEF

B

C

A

F

D

E

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 6/21

Theorem 4.5: Angle-Angle-Side

(AAS) Congruence TheoremYou are given that two angles of 

¨ABC are congruent to twoangles of ¨DEF. By the Third

Angles Theorem, the thirdangles are also congruent.That is, �B $ �E. Notice thatBC is the side includedbetween �B and �C, and EFis the side included between�E and �F. You can applythe ASA CongruencePostulate to conclude that¨ABC $ ¨DEF.

B

C

A

F

D

E

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 7/21

Ex. 1 Developing Proof Is it possible to prove

the triangles are

congruent? If so,state the postulate or

theorem you would

use. Explain your

reasoning.

G

E

JF

H

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 8/21

Ex. 1 Developing Proof A. In addition to the angles

and segments that aremarked, �EGF $�JGH

by the Vertical AnglesTheorem. Two pairs of corresponding anglesand one pair of corresponding sides are

congruent. You can usethe AAS CongruenceTheorem to prove that¨EFG $ ¨JHG.

G

E

JF

H

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 9/21

Ex. 1 Developing Proof Is it possible to prove

the triangles are

congruent? If so,state the postulate or

theorem you would

use. Explain your

reasoning.

N

M

Q

P

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 10/21

Ex. 1 Developing Proof B. In addition to the

congruent segments

that are marked, NP$ NP. Two pairs of corresponding sidesare congruent. Thisis not enough

information to provethe triangles arecongruent.

N

M

Q

P

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 11/21

Ex. 1 Developing Proof Is it possible to prove

the triangles are

congruent? If so,state the postulate or

theorem you would

use. Explain your

reasoning.

UZ WX AND UWWX.

U

W

Z

X

1

2

3

4

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 12/21

Ex. 1 Developing Proof The two pairs of 

parallel sides can be

used to show�

1$

�3 and �2 $ �4.Because the includedside WZ is congruentto itself, ¨WUZ $

¨ZXW by the ASACongruencePostulate.

U

W

Z

X

1

2

3

4

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 13/21

Ex. 2 Proving Triangles are

CongruentGiven: AD EC, BD $ BCProve: ¨ABD $ ¨EBC

Plan for proof: Notice that�ABD and �EBC are

congruent. You are

given that BD $ BC

. Use the fact that AD ECto identify a pair of congruent angles.

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 14/21

Proof:Statements:

1. BD $ BC

2. AD EC3. �D $ �C4. �ABD $ �EBC

5. ¨ABD $ ¨EBC

R easons:

1.

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 15/21

Proof:Statements:

1. BD $ BC

2. AD EC3. �D $ �C4. �ABD $ �EBC

5. ¨ABD $ ¨EBC

R easons:

1. Given

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 16/21

Proof:Statements:

1. BD $ BC

2. AD EC3. �D $ �C4. �ABD $ �EBC

5. ¨ABD $ ¨EBC

R easons:

1. Given

2. Given

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 17/21

Proof:Statements:

1. BD $ BC

2. AD EC3. �D $ �C4. �ABD $ �EBC

5. ¨ABD $ ¨EBC

R easons:

1. Given

2. Given

3. Alternate Interior

Angles

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 18/21

Proof:Statements:

1. BD $ BC

2. AD EC3. �D $ �C4. �ABD $ �EBC

5. ¨ABD $ ¨EBC

R easons:

1. Given

2. Given

3. Alternate Interior

Angles

4. Vertical AnglesTheorem

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 19/21

Proof:Statements:

1. BD $ BC

2. AD EC3. �D $ �C4. �ABD $ �EBC5. ¨ABD $ ¨EBC

R easons:

1. Given

2. Given3. Alternate Interior

Angles

4. Vertical Angles

Theorem5. ASA Congruence

Theorem

B

A

ED

C

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 20/21

Note:� You can often use more than one method

to prove a statement. In Example 2, you

can use the parallel segments to show

that �D $ �C and �A $ �E. Then you

can use the AAS Congruence Theorem to

prove that the triangles are congruent.

8/7/2019 4.4 ASA AND AAS

http://slidepdf.com/reader/full/44-asa-and-aas 21/21

Done by:

� Wessam and Hadi

�Grade 7A