Warm Up 5.3 on desk Do the Daily Quiz 5.2

5.3 ESSENTIAL QUESTION How are triangles congruent using ASA and AAS postulates?

Angle-Side-Angle Congruence Postulate (ASA) 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 two triangles are congruent.

Use the ASA Congruence Postulate Then, ∆ABC  ∆DFE. SOLUTION Example 1 Determine When To Use ASA Congruence a. b. Use the ASA Congruence Postulate Then, ∆ABC  ∆DFE. SOLUTION a. C  E, B  F, and BC  FE. R  Y and S  X. b. RT  YZ, but are not included between the congruent angles, so you cannot use the ASA Congruence Postulate. 4

Angle-Angle-Side Congruence Theorem (AAS) 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 two triangles are congruent. Angle-Angle-Side Congruence Theorem (AAS)

What information is needed to show that ∆JKL  ∆NML? Example 2 Determine What Information is Missing What information is needed to show that ∆JKL  ∆NML? (by AAS Congruence Theorem) SOLUTION You are given KL  ML. Because KLJ and MLN are vertical angles, KLJ  MLN. We need to know that J  N. 6

Example 2 Determine What Information is Missing 7

Yes, we can use the AAS Congruence Postulate and that ∆EFG  ∆JHG. Example 3 Decide Whether Triangles are Congruent Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. a. SOLUTION a. We know that: S EF  JH A E  J A FGE  HGJ Yes, we can use the AAS Congruence Postulate and that ∆EFG  ∆JHG. 8

We know only that MP  QN and NP  NP. Example 3 Decide Whether Triangles are Congruent b. We know only that MP  QN and NP  NP. You cannot use AAS or ASA! Neither SSS or SAS! Because sides aren’t parallel: 9

Since sides are parallel: Example 3 Decide Whether Triangles are Congruent c. Since sides are parallel: A UZW  XWZ alternate interior angles c. S WZ  WZ A UWZ  XZW alternate interior angles Use the ASA Congruence Postulate to conclude that ∆WUZ  ∆ZXW. 10

Alternate Interior Angles Theorem Vertical Angles Theorem 4. Example 4 Prove Triangles are Congruent A step in the Cat’s Cradle string game creates the triangles shown. Prove that ∆ABD  ∆EBC. A D B C E SOLUTION BD  BC, AD || EC ∆ABD  ∆EBC Statements Reasons Given 1. BD  BC Given 2. AD || EC D  C 3. Alternate Interior Angles Theorem Vertical Angles Theorem 4. ABD  EBC ASA Congruence Postulate 5. ∆ABD  ∆EBC 11

Checkpoint Decide Whether Triangles are Congruent Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. 1. 2. 3. ANSWER yes; AAS Congruence Theorem ANSWER no ANSWER no

Classwork 5.3A IXL: continue Skills B4 & B5