Triangle Congruence by ASA and AAS February 27, 2012

Warm-up Practice 4-2: Workbook p. 42, #1-15

Warm-up

Warm-up

Questions on Homework?

Questions on Homework? #33: Statements Reasons

Questions on Homework? 41.Prove ΔFGK ≅ ΔKLF 42. Prove ΔACB ≅ ΔECD and bisect each other 43. Prove ΔGJK ≅ ΔGMK bisects ∠JGM 44. Prove ΔAMC ≅ ΔMBD ┴ ; ┴ ; M is midpoint of

Section 4-3 Triangle Congruence by ASA and AAS Objectives: Today you will learn to prove triangles congruent using the ASA Postulate and AAS Theorem Remember: Quiz on Wednesday on 4-1, 4-2, and 4-3

Investigation Congruence Applet: http://illuminations.nctm.org/ActivityDetail.aspx?ID=4 Each Triangle Congruence Postulate uses three elements (sides and angles) to prove congruence. Today let’s investigate using Angles (2 or 3) and/or one Side

Conclusions Angle-Side-Angle (ASA) Postulate In which configuration(s) were the triangles always congruent? Angle-Side-Angle (ASA) Postulate Angle-Angle-Side (AAS) Theorem In which configuration(s) were the triangles sometimes congruent? Angle-Angle-Angle (AAA) – can’t use!

Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. ΔTUV ≅ ΔWXY

Angle-Angle-Side (AAS) Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. ΔABC ≅ ΔDEF

Angle-Angle-Side (AAS) Theorem Proof: ∠C ≅ ∠F Given ∠B and ∠E are right ∠’s Given ∠B ≅ ∠E Rt ∠’s ≅ ∠A ≅ ∠D Thm 4.1 ΔABC ≅ ΔDEF ASA Postulate ΔABC ≅ ΔDEF

SSS Postulate SAS Postulate ASA Postulate AAS Theorem Four Ways to Prove Δ’s ≅ SSS Postulate SAS Postulate ASA Postulate AAS Theorem

Which Postulate/Thm? (if possible) and write Congruency Statement

Example 1: Prove: ΔAXP ≅ ΔBYP Given: ≅ segments and angles as marked

Example 2: Prove ΔABC ≅ ΔCDA Given: ≅ angles and || segments as marked

Example 3: Prove ΔABE ≅ ΔCDE

Example 4: Prove ΔJKL ≅ ΔPML

Example 5: Prove ΔQRT ≅ ΔSTR

Example 6: Find the values for x and y Given: ΔABD ≅ ΔACD

Example 7: Find the values for x and y Given: ΔABD ≅ ΔACD If BC = 6, BD = ____ If m∠C = 55, m∠B = _____ and m∠BAC = _____ If m∠BAD = 40, m∠B = ____

Wrap-up Today you learned to prove triangles congruent using the ASA and AAS Postulates Tomorrow you’ll learn about the HL Theorem and about and how to use CPCTC. Homework: pp. 197 – 199: 1 – 29, 31-34: write the proof if you can deduce the conclusion from the given Quiz on Wednesday on 4-1, 4-2, and 4-3