Triangle Congruence: Angle-Side-Angle & Angle-Angle-Side ASA & AAS

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Presentation transcript:

Triangle Congruence: Angle-Side-Angle & Angle-Angle-Side ASA & AAS Lesson 30 Triangle Congruence: Angle-Side-Angle & Angle-Angle-Side ASA & AAS

Review Vocabulary The angle formed by two adjacent sides of a polygon is called an included angle. What is the included angle of 𝐴𝐷 & 𝐴𝐵 ? ∠A The common side of two consecutive angles of a polygon is called an included side. What is the included side of ∠D & ∠C? 𝐷𝐶

Postulate 16: -Angle-Side-Angle (ASA) Triangle Congruence Postulate If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent. ΔKGB ≅ ΔCIA, ASA

Find the value of y ∠FMH & ∠JML are vertical angles so they are congruent. Therefore ΔHMF ≅ ΔJML, ASA By CPCTC 𝐹𝐻 ≅ 𝐿𝐽 2𝑦−5=63 2𝑦=68 𝑦=34

What can you conclude about ∠G and ∠I? Hint: Corollary 18-1-1 (No Choice Thm) ∠G ≅ ∠I Now what can you conclude about the 2 triangles? They are congruent by ASA This leads us to our next theorem and 4th triangle congruence

Theorem 30-1: Angle-Angle-Side (AAS) If 2 angles and the nonincluded side of one triangle are congruent to 2 angles and the nonincluded side of another triangle, then the triangles are congruent. ΔKGB ≅ ΔCIA, AAS

GIVEN: ∠𝐿 ≅∠𝑁 𝑀𝑃 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐿𝑀𝑁 PROVE: ΔMPL ≅ ΔMPN Statements Reasons ∠𝐿 ≅∠𝑁 𝑀𝑃 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐿𝑀𝑁 ∠𝐿𝑀𝑃 ≅∠𝑁𝑀𝑃 𝑀𝑃 ≅ 𝑀𝑃 ΔMPL ≅ ΔMPN Given Def. of Angle Bisector Reflexive of ≅ AAS

GIVEN: 𝑄𝑇 ⊥ 𝑅𝑆 ; 𝑄𝑇 bisects ∠𝑅𝑄𝑆 PROVE: ΔRTQ ≅ ΔSTQ Statements Reasons 𝑄𝑇 ⊥ 𝑅𝑆 ∠𝑄𝑇𝑅≅∠𝑄𝑇𝑆 𝑄𝑇 bisects ∠𝑅𝑄𝑆 ∠𝑇𝑄𝑅≅∠𝑇𝑄𝑆 𝑄𝑇 ≅ 𝑄𝑇 ΔRTQ ≅ ΔSTQ Given Thm 5-4 (⊥ lines, form ≅ ∠’s) Def. of Angle Bisector Reflexive of ≅ ASA

Conclusion Proving triangles will prepare you for : Lesson 38: Perpendicular and Angle Bisectors of Triangles Lesson 46: Triangle Similarity Lesson 51: Properties of Isosceles and Equilateral Triangles Lesson 55: Triangle Midsegment Thm