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4.4 Proving Triangles are Congruent: ASA and AAS. Geometry Mrs. Spitz Fall 2004. Objectives:. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem Use congruence postulates and theorems in real-life problems. Assignment.

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Objectives
Objectives:

  • Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem

  • Use congruence postulates and theorems in real-life problems.


Assignment
Assignment

  • 4.4 pp. 223-225 #1-22 all

  • Quiz after this section


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.

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


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.

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


Theorem 4 5 angle angle side aas congruence theorem1

Given: congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.A  D, C  F, BC  EF

Prove: ∆ABC  ∆DEF

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


Theorem 4 5 angle angle side aas congruence theorem2

You are given that two angles of congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.∆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.

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


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.

Ex. 1 Developing Proof


Ex 1 developing proof1

A. In addition to the angles and segments that are marked, state the postulate or theorem you would use. Explain your reasoning.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.

Ex. 1 Developing Proof


Ex 1 developing proof2

Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof


Ex 1 developing proof3

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.

Ex. 1 Developing Proof


Ex 1 developing proof4

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.

Ex. 1 Developing Proof

1

2

3

4


Ex 1 developing proof5

The two pairs of parallel sides can be used to show state the postulate or theorem you would use. Explain your reasoning.1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate.

Ex. 1 Developing Proof

1

2

3

4


Ex 2 proving triangles are congruent

Given: AD state the postulate or theorem you would use. Explain your reasoning.║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.

Ex. 2 Proving Triangles are Congruent


Proof

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1.

Proof:


Proof1

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1. Given

Proof:


Proof2

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Proof:


Proof3

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Proof:


Proof4

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

Proof:


Proof5

Statements: state the postulate or theorem you would use. Explain your reasoning.

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

ASA Congruence Theorem

Proof:


Note: state the postulate or theorem you would use. Explain your reasoning.

  • 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.


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