4.5 Proving 4.5 Proving ΔΔs are s are : : ASA and AASASA and AAS

Objectives:Objectives:

• Use the ASA Postulate to prove triangles Use the ASA Postulate to prove triangles congruentcongruent

• Use the AAS Theorem to prove triangles Use the AAS Theorem to prove triangles congruentcongruent

Postulate 4.3 Postulate 4.3 ((ASAASA):):Angle-Side-Angle Congruence Angle-Side-Angle Congruence

PostulatePostulate

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

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

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

Proof of the Angle-Angle-Side Proof of the Angle-Angle-Side (AAS) Congruence Theorem(AAS) Congruence Theorem

Given: Given: A A D, D, C C F, BC F, BC EF EF

Prove: Prove: ∆ABC ∆ABC ∆DEF ∆DEF

Paragraph ProofParagraph Proof

You are given that two angles of You are given that two angles of ∆ABC are congruent to two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, congruent. That is, B B E. Notice that BC is the side included E. Notice that BC is the side included between between B and B and C, and EF is the side included between C, and EF is the side included between E and E and F. F. You can apply the ASA Congruence Postulate to conclude that ∆ABC You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF. ∆DEF.

A B

C

D

E

F

Example 1:Example 1:

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

Example 1:Example 1:

In addition to the In addition to the angles and segments angles and segments that are marked, that are marked, EGF EGF JGH by the JGH by the Vertical Angles Vertical Angles Theorem. Two pairs Theorem. Two pairs of corresponding of corresponding angles and one pair of angles and one pair of corresponding sides corresponding sides are congruent. Thus, are congruent. Thus, you can use the you can use the AAS AAS Congruence Congruence TheoremTheorem to prove that to prove that ∆EFG ∆EFG ∆JHG. ∆JHG.

Example 2:Example 2:

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

Example 2:Example 2:

In addition to the In addition to the congruent segments congruent segments that are marked, NP that are marked, NP NP. Two pairs of NP. Two pairs of corresponding sides corresponding sides are congruent. This are congruent. This is is not enough not enough informationinformation to prove to prove the triangles are the triangles are congruent. congruent.

Example 3:Example 3:

Given: ADGiven: AD║EC, BD ║EC, BD BC BC

Prove: Prove: ∆ABD ∆ABD ∆EBC ∆EBC

Plan for proof: Notice that Plan for proof: Notice that ABD and ABD and EBC are EBC are congruent. You are given congruent. You are given that that BD BD BC. BC. Use the fact Use the fact that that AD AD ║EC to identify a ║EC to identify a pair of congruent angles.pair of congruent angles.

Proof:Proof:

Statements:Statements:1.1. BD BD BC BC2.2. AD AD ║ EC║ EC3.3. D D CC

4.4. ABD ABD EBCEBC

5.5. ∆∆ABD ABD ∆EBC ∆EBC

Reasons:Reasons:

1.1. GivenGiven

2.2. GivenGiven

3.3. If || lines, then alt. If || lines, then alt. int. int. s are s are

4.4. Vertical Angles Vertical Angles TheoremTheorem

5.5. ASA Congruence ASA Congruence PostulatePostulate

AssignmentAssignment

Geometry:Geometry:Pg. 210 #4, 6, 10, 12, 25 – 28Pg. 210 #4, 6, 10, 12, 25 – 28

Pre-AP Geometry:Pre-AP Geometry:Pg. 211 #10 – 20 evens, #25 - Pg. 211 #10 – 20 evens, #25 -

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