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4-5 Objective: Prove Triangles Congruent by ASA and AAS Quick Review over SAS and HL ASA- Angle Side Angle • Angle Side Angle Congruence Postulate. If two angles and the included side of two angles are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. – Notice the side is in between the angles – See Page 249 AAS- Angle Angle Side • 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. – Has to go in order around the triangle: Angle Angle Side – See Page 249 Important to Note • AAA is NOT a characteristic to show that two triangles are congruent. EXAMPLE 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. SOLUTION a. The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. The triangles are congruent by the AAS Congruence Theorem. Extra Example 1 Flow Proofs • Making a proof like a flow chart • Use arrows to show the flow of a logical argument. – Still start with given • Use arrows to show what you can conclude from the given – Still need reasons • Write below the flow chart boxes EXAMPLE 2 Prove the AAS Congruence Theorem Prove the Angle-Angle-Side Congruence Theorem. Write a proof. GIVEN PROVE A D, ABC C DEF F, BC EF GUIDED PRACTICE 1. for Examples 1 and 2 In the diagram at the right, what postulate or theorem can you use to RST VUT ? Explain. prove that SOLUTION STATEMENTS REASONS S U Given RS UV Given RTS UTV The vertical angles are congruent GUIDED PRACTICE for Examples 1 and 2 ANSWER AAS; RTS UTV because they are vertical angles. EXAMPLE 3 Write a flow proof In the diagram, CE BD and CAB Write a flow proof to show GIVEN PROVE CE BD, CAB ABE ADE ABE CAD CAD. ADE GUIDED PRACTICE 3. for Examples 3 and 4 In Example 3, suppose ABE ADE is also given. What theorem or postulate besides ASA ABE ADE? can you use to prove that ANSWER AAS Congruence Theorem. SSS, SAS, HL, ASA, and AAS • Top of page 252 Daily Homework Quiz For use after Lesson 4.5 Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 1. Daily Homework Quiz For use after Lesson 4.5 Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 2. Daily Homework Quiz Write flow proof. Given : BD bisects ABC, Prove : ABD CBD 3. For use after Lesson 4.5 A C Daily Homework Quiz For use after Lesson 4.5 Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 1. ANSWER ASA . Daily Homework Quiz For use after Lesson 4.5 Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 2. ANSWER not necessarily congruent . Daily Homework Quiz Write flow proof. Given : BD bisects ABC, Prove : ABD CBD 3. For use after Lesson 4.5 A C Daily Homework Quiz ANSWER For use after Lesson 4.5 Homework • 3-10, 14 –21, 23 – 25, 31 – 34, 41 - 43