Proving Triangles are Congruent

SSS, SAS; ASA; AASCCSS: 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 CONGRUENCE POSTULATES

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

and thenIfSides are congruent

1. AB DE

2. BC EF

3. AC DF

Angles are congruent

4. A D

5. B E

6. C F

Triangles are congruent

ABC DEF

SSS AND SAS CONGRUENCE POSTULATES

POSTULATE

POSTULATE 19 Side - Side - Side (SSS) Congruence Postulate

Side MN QR

Side PM SQ

Side NP RS

If

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

then MNP QRSS

S

S

SSS AND SAS CONGRUENCE POSTULATES

The SSS Congruence Postulate is a shortcut for provingtwo triangles are congruent without using all six pairsof corresponding parts.

POSTULATE

SSS AND SAS CONGRUENCE POSTULATES

POSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate

Side PQ WX

Side QS XY

then PQS WXYAngle Q X

If

If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

A

S

S

Congruent Triangles in a Coordinate Plane

AC FH

AB FGAB = 5 and FG = 5

SOLUTION

Use the SSS Congruence Postulate to show that ABC FGH.

AC = 3 and FH = 3

Congruent Triangles in a Coordinate Plane

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 3 2 + 5

2

= 34

BC = (– 4 – (– 7)) 2 + (5 – 0 )

2

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 5 2 + 3

2

= 34

GH = (6 – 1) 2 + (5 – 2 )

2

Use the distance formula to find lengths BC and GH.

Congruent Triangles in a Coordinate Plane

BC GH

All three pairs of corresponding sides are congruent, ABC FGH by the SSS Congruence Postulate.

BC = 34 and GH = 34

SSS postulate SAS postulate

T C

S G

The vertex of the included angle is the point in common.

SSS postulateSAS postulate

SSS postulate

Not enough info

SSS postulateSAS postulate

Not Enough InfoSAS postulate

SSS postulate

Not Enough Info

SAS postulate SAS postulate

Congruent Triangles in a Coordinate Plane

MN DE

PM FEPM = 5 and FE = 5

SOLUTION

Use the SSS Congruence Postulate to show that NMP DEF.

MN = 4 and DE = 4

Congruent Triangles in a Coordinate Plane

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 4 2 + 5

2

= 41

PN = (– 1 – (– 5)) 2 + (6 – 1 )

2

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= (-4) 2 + 5

2

= 41

FD = (2 – 6) 2 + (6 – 1 )

2

Use the distance formula to find lengths PN and FD.

Congruent Triangles in a Coordinate Plane

PN FD

All three pairs of corresponding sides are congruent, NMP DEF by the SSS Congruence Postulate.

PN = 41 and FD = 41

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.

B

C

A

F

D

E

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.

B

C

A

F

D

E

Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremGiven: A D, C

F, BC EF

Prove: ∆ABC ∆DEF

B

C

A

F

D

E

Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremYou 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, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF.

B

C

A

F

D

E

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

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.

G

E

JF

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.

N

M

Q

P

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.

N

M

Q

P

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 UW

║WX.

U

W

Z

X

12

34

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.

U

W

Z

X

12

34

Ex. 2 Proving Triangles are CongruentGiven: AD ║EC, BD

BC

Prove: ∆ABD ∆EBC

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.

B

A

ED

C

Proof:

Statements:

1. BD BC

2. AD ║ EC

3. D C

4. ABD EBC

5. ∆ABD ∆EBC

Reasons:

1.

B

A

ED

C

Proof:

Statements:

1. BD BC

2. AD ║ EC

3. D C

4. ABD EBC

5. ∆ABD ∆EBC

Reasons:

1. Given

B

A

ED

C

Proof:

Statements:

1. BD BC

2. AD ║ EC

3. D C

4. ABD EBC

5. ∆ABD ∆EBC

Reasons:

1. Given

2. Given

B

A

ED

C

Proof:

Statements:

1. BD BC

2. AD ║ EC

3. D C

4. ABD EBC

5. ∆ABD ∆EBC

Reasons:

1. Given

2. Given

3. Alternate Interior Angles

B

A

ED

C

Proof:

Statements:

1. BD BC

2. AD ║ EC

3. D C

4. ABD EBC

5. ∆ABD ∆EBC

Reasons:

1. Given

2. Given

3. Alternate Interior Angles

4. Vertical Angles Theorem

B

A

ED

C

Proof:

Statements:1. BD BC2. AD ║ EC3. D C4. ABD EBC5. ∆ABD ∆EBC

Reasons:1. Given2. Given3. Alternate Interior

Angles4. Vertical Angles

Theorem5. ASA Congruence

Theorem

B

A

ED

C

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.