5.3 Proving Triangles are Congruent – ASA & AAS

Objectives: Show triangles are congruent using ASA and AAS.

Key Vocabulary Included Side

Postulates 14 Angle–Side–Angle (ASA) Congruence Postulate

Theorems 5.1 Angle-Angle-Side (AAS) Congruence Theorem

Definition: Included Side An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

Example: Included Sided B C X Z Y The side between 2 angles INCLUDED SIDE

Postulate 14 (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.

Two angles and the INCLUDED side Angle-Side-Angle (ASA) Congruence Postulate A S A S Two angles and the INCLUDED side

You are given that C  E, B  F, and BC  FE. Example 1 Determine When To Use ASA Congruence Based on the diagram, can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain your reasoning. a. b. You can use the ASA Congruence Postulate to show that ∆ABC  ∆DFE. SOLUTION a. You are given that C  E, B  F, and BC  FE. You are given that R  Y and S  X. b. You know that RT  YZ, but these sides are not included between the congruent angles, so you cannot use the ASA Congruence Postulate. 10

Example 2: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent. Explain. Two congruent angle pairs are given, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent.

Your Turn Determine if you can use ASA to prove NKL  LMN. Explain. By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied.

Theorem 5.1 (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.

AAS A D B F C E OR X H Y Z I J

Angle-Angle-Side (AAS) Congruence Theorem Two Angles and One Side that is NOT included

Example 3 Determine What Information is Missing What additional congruence is needed to show that ∆JKL  ∆NML by the AAS Congruence Theorem? SOLUTION You are given KL  ML. Because KLJ and MLN are vertical angles, KLJ  MLN. The angles that make KL and ML the non-included sides are J and N, so you need to know that J  N. 16

Use the AAS Congruence Theorem to conclude that ∆EFG  ∆JHG. Example 4 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. b. c. a. SOLUTION Given a. EF  JH E  J Given FGE  HGJ Vertical Angles Theorem Use the AAS Congruence Theorem to conclude that ∆EFG  ∆JHG. 17

Use the ASA Congruence Postulate to conclude that ∆WUZ  ∆ZXW. Example 4 Decide Whether Triangles are Congruent b. c. b. Based on the diagram, you know only that MP  QN and NP  NP. You cannot conclude that the triangles are congruent. UZW  XWZ Alternate Interior Angles Theorem c. Reflexive Prop. of Congruence WZ  WZ Alternate Interior Angles Theorem UWZ  XZW Use the ASA Congruence Postulate to conclude that ∆WUZ  ∆ZXW. 18

Example 5: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 6: 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 7: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 8: In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information (CBD) to prove the triangles are congruent.

BD  BC, AD || EC ∆ABD  ∆EBC Prove Triangles are Congruent Example 9 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

Your Turn: Complete the statement: You can use the ASA Congruence Postulate when the congruent sides are between the corresponding congruent angles. _____ ? 1. ANSWER included Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. 2. 3. 4. ANSWER yes; AAS Congruence Theorem ANSWER no ANSWER no

Ways To Prove Triangles Are Congruent Congruence Shortcuts SSS SAS ASA AAS Ways To Prove Triangles Are Congruent

Congruence Shortcuts

AAA and SSA??? Does AAA and SSA provide enough information to determine the exact shape and size of a triangle?

AAA and SSA??? Does AAA and SSA provide enough information to determine the exact shape and size of a triangle? NO

Not Congruence Shortcuts NO BAD WORDS SSA AAA Do Not prove Triangle Congruence NO CAR INSURANCE

Triangle Congruence Practice Your Turn

Is it possible to prove the Δs are ? ( )) ) )) ) (( (( ( No, there is no AAA ! CBD Yes, ASA

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). G I H J K ΔGIH  ΔJIK by AAS

DEF NLM by ____ ASA

What other pair of angles needs to be marked so that the two triangles are congruent by AAS?

What other pair of angles needs to be marked so that the two triangles are congruent by ASA?

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). E A C B D ΔACB  ΔECD by SAS

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). J K L M ΔJMK  ΔLKM by SAS or ASA

Cannot Be Determined (CBD) Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). J T L K V U Cannot Be Determined (CBD)

Cannot Be Determined (CBD) – SSA is not a valid Congruence Shortcut.

Yes, ∆TNS ≅ ∆UHS by AAS

Review SSS, SAS, ASA, and AAS use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Remember!

Example 10: Using CPCTC A and B are on the edges of a ravine. What is AB? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so AB = 18 mi.

Your Turn A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so JK = 41 ft.

Joke Time Which one came first the egg or the chicken? I don't care I just want my breakfast served. What do you call a handsome intelligent sensitive man? A rumor. What does a clock do when it's hungry? Goes back 4 secounds!!!

Assignment Pg. 253 - 256 #1 – 21 odd, 25 – 45 odd