4.5 Proving Δ s are  : ASA and AAS

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4.5 Proving Δ s are  : ASA and AAS. Objectives:. Use the ASA Postulate to prove triangles congruent Use the AAS Theorem to prove triangles congruent.

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### 4.5 Proving Δs are  : ASA and AAS

Objectives:
• Use the ASA Postulate to prove triangles congruent
• Use the AAS Theorem to prove triangles congruent
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 4.3 (ASA):Angle-Side-Angle Congruence Postulate
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 (AAS): Angle-Angle-Side Congruence Theorem
Proof of the Angle-Angle-Side (AAS) Congruence Theorem

Given: A  D, C  F, BC  EF

Prove: ∆ABC  ∆DEF

D

A

B

F

C

Paragraph Proof

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, 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.

E

Example 1:

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

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. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.Example 1:
Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.Example 2:
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. Example 2:

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.

Example 3:
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

If || lines, then alt. int. s are 

Vertical Angles Theorem

ASA Congruence Postulate

Proof:
Assignment
• Geometry: Pg. 210 #4, 6, 10, 12, 25 – 28
• Pre-AP Geometry: Pg. 211 #10 – 20 evens, #25 - 28