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Transcript
Prove Triangles Congruent by ASA & AAS Lesson 4.10 Use two more methods to prove congruences Vocabulary A flow proof uses arrows to show the flow of a logical argument. ASA Congruence Postulate: If two angles and the included side of one triangle are congruent to two angles & included side of a second triangle, then the two triangles are congruent AAS Congruence Theorem: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. Triangle Congruence Review Postulates we can use CAN’T USE Warm-Up Exercises Lesson 4.5, For use with pages 249-255 Tell whether the pair of triangles is congruent or not and why. ANSWER Yes; HL Thm. 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. 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. EXAMPLE 1 Identify congruent triangles b. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent. c. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate. 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 ASA Congruence Postulate AAS Congruence Theorem 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 RST UTV VUT The vertical angles are congruent AAS for Examples 1 and 2 GUIDED PRACTICE Rewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof. 2. ABC GIVEN PROVE m 1+m 2+m 3 = 180° STATEMENTS 1. Draw BD parallel to AC . 2. m 4 + m 2 + m 5 = 180° REASONS 1. Parallel Postulate 2. Angle Addition Postulate and definition of straight angle 3. 1 4, 3 4. m 1= m 4,m 5. m 1+m 2+m 3. Alternate Interior Angles 5 3= m 5 3 = 180° Theorem 4. Definition of congruent angles 5. Substitution Property of Equality 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