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

Postulate 4.3 ( ASA ): Angle-Side-Angle 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.

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

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

Given:  A   D,  C   F, BC  Prove: ∆ABC  ∆DEF EF

D A B F C

Paragraph Proof

E

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

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 1:

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 2:

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

the triangles are congruent. to prove

Example 3:

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.

Proof:

Statements: 1.

BD  BC 2.

3.

AD ║ EC  D   C 4.

 ABD   EBC 5.

∆ABD  ∆EBC Reasons: 1.

2.

3.

Given Given If || lines, then alt. int.  s are  4.

5.

Vertical Angles Theorem ASA Congruence Postulate

Assignment

Geometry: Pg. 210 #9 – 14, 25 – 28 Pre-AP Geometry: Pg. 211 #10 – 18 evens, #25 - 28