Proving Triangles are Congruent SSS, SAS; ASA; AAS

CCSS: G.CO7

Standards for Mathematical Practices

• • • • • • • •

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

CCSS:G.CO 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.

ESSENTIAL QUESTION

• •

How do we show that triangles are congruent? How do we use triangle congruence to plane and write proves ,and prove that constructions are valid?

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.

Proving Triangles are Congruent: SSS and SAS

SSS AND SAS C ONGRUENCE P OSTULATES

If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

If 1.

2.

3.

Sides are congruent

AB BC AC DE EF DF

and Angles are congruent 4.

5.

B

6.

A C D E F

then Triangles are congruent



ABC



DEF

SSS AND SAS C ONGRUENCE P OSTULATES POSTULATE POSTULATE 19 Side - Side - Side (SSS) Congruence Postulate

If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

If

S S

MN

S S

NP

S S

PM QR RS SQ

then 

MNP



QRS

SSS AND SAS C ONGRUENCE P OSTULATES

The SSS Congruence Postulate is a shortcut for proving two triangles are congruent without using all six pairs of corresponding parts.

SSS AND SAS C ONGRUENCE P OSTULATES POSTULATE POSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate

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

S S A A S S

PQ Q QS WX X XY

then 

PQS



WXY

Congruent Triangles in a Coordinate Plane

Use the SSS Congruence Postulate to show that 

ABC

 

FGH

.

S OLUTION

AC

= 3 and

FH

= 3

AB

= 5 and

FG

= 5

AC



AB



FH FG

Congruent Triangles in a Coordinate Plane

Use the distance formula to find lengths

BC

and

GH

.

d

= (

x

2

–

x

1

)

2

+

(

y

2

–

y

1 ) 2

BC

= (

– 4

– (

– 7

))

2

+

( 5

–

0 ) 2

= 3

2

+ 5 = 34

2

d

= (

x

2

–

x

1

)

2

+

(

y

2

–

y

1 ) 2

GH

= (

6

–

1

)

2

+

( 5

–

2 ) 2

= 5

2

+ 3

2

= 34

Congruent Triangles in a Coordinate Plane

BC

= 34 and

GH

= 34

BC



GH

All three pairs of corresponding sides are congruent, 

ABC

 

FGH

by the SSS Congruence Postulate.

SSS  postulate SAS  postulate

 T  C  S  G The vertex of the included angle is the point in common.

SAS  postulate SSS  postulate

SSS  postulate Not enough info

SSS  postulate SAS  postulate

Not Enough Info SAS  postulate

SSS  postulate Not Enough Info

SAS  postulate SAS  postulate

Congruent Triangles in a Coordinate Plane

Use the SSS Congruence Postulate to show that 

NMP

 

DEF

.

S OLUTION

MN

= 4 and

DE

= 4

PM

= 5 and

FE

= 5

MN PM



DE



FE

Congruent Triangles in a Coordinate Plane

Use the distance formula to find lengths

PN

and

FD

.

d

= (

x

2

–

x

1

)

2

+

(

y

2

–

y

1 ) 2

PN

= (

– 1

– (

– 5

))

2

+

( 6

–

1 ) 2

= 4

2

+ 5 = 41

2

d

= (

x

2

–

x

1

)

2

+

(

y

2

–

y

1 ) 2

FD

= (

2

–

6

)

2

+

( 6

–

1 ) 2

= (-4)

2

+ 5

2

= 41

Congruent Triangles in a Coordinate Plane

PN

= 41 and

FD

= 41

PN



FD

All three pairs of corresponding sides are congruent, 

NMP

 

DEF

by the SSS Congruence Postulate.

Proving Triangles are Congruent ASA; AAS

Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate

•

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

C B D E A F

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

•

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

C B D E A F

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

Given:



A

 

D,



C

 

F, BC



EF Prove: ∆ABC



∆DEF

B A E C F D

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, between



E and

  

B

 

E. Notice that BC is the side included B and



C, and EF is the side included between F. You can apply the ASA Congruence Postulate to conclude that ∆ABC



∆DEF.

C B D E A F

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.

F E G J H

Ex. 1 Developing Proof

A. In addition to the angles and segments that are marked,



EGF



JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG



∆JHG.

F E G J H

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.

M N P Q

Ex. 1 Developing Proof

B. In addition to the congruent segments that are marked, NP



NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

M N P Q

Ex. 1 Developing Proof

U

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 UW ║WX.

W 3 4 X 1 2 Z

Ex. 1 Developing Proof

The two pairs of parallel sides can be used to show



1

 

3 and



2

 

4. Because the included side WZ is congruent to itself, ∆WUZ



∆ZXW by the ASA Congruence Postulate.

W 3 4 U X 1 2 Z

Ex. 2 Proving Triangles are Congruent

Given: AD ║EC, BD



Prove: ∆ABD



∆EBC BC Plan for proof: Notice that



ABD and



EBC are congruent. You are given that BD



BC . Use the fact that AD ║EC to identify a pair of congruent angles.

A D B E C

C A

Proof:

D

Statements: 1. BD



BC 2. AD ║ EC 3.

4.

5.



D

 

C



ABD

 

EBC ∆ABD



∆EBC

B E

Reasons: 1.

C A

Proof:

D

Statements: 1. BD



BC 2. AD ║ EC 3.

4.

5.



D

 

C



ABD

 

EBC ∆ABD



∆EBC

B E

Reasons: 1. Given

C A

Proof:

D

Statements: 1. BD



BC 2. AD ║ EC 3.

4.

5.



D

 

C



ABD

 

EBC ∆ABD



∆EBC

B E

Reasons: 1. Given 2. Given

C A

Proof:

D

Statements: 1. BD



BC 2. AD ║ EC 3.

4.

5.



D

 

C



ABD

 

EBC ∆ABD



∆EBC

B E

Reasons: 1. Given 2. Given 3. Alternate Interior Angles

C A

Proof:

D

Statements: 1. BD



BC 2. AD ║ EC 3.

4.

5.



D

 

C



ABD

 

EBC ∆ABD



∆EBC

B E

Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem

C A

Proof:

D

Statements: 1. BD



BC 2. AD ║ EC 3.



D

 

C 4.

5.



ABD

 

EBC ∆ABD



∆EBC

B E

Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem 5. ASA Congruence Theorem

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.