4.3 and 4.4 Proving Δs are  : SSS and SAS AAS and ASA

Objectives: Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem and the SAS and SSS. Use congruence postulates and theorems in real-life problems.

Postulate 21: Angle-Side-Angle (ASA) 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.

Meaning: A Side-Side-Side (SSS) ___ ___ If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are . B C ___ Meaning: E If seg AB  seg ED, seg AC  seg EF seg BC  seg DF, then ΔABC  ΔEDF. ___ ___ D ___ F

Theorem 4.5: Angle-Angle-Side (AAS) 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.

Post. 20 Side-Angle-Side post. (SAS) If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

If seg BC  seg YX, seg AC  seg ZX, and C  X, then ΔABC  ΔZXY. ) ( C A X Z

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF

Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW║WX. 1 2 3 4

Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC . Use the fact that AD ║EC to identify a pair of congruent angles.

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1.

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Given

Proof: Statements: Reasons: BD  BC Given AD ║ EC D  C ABD  EBC

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Alternate Interior Angles

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Alternate Interior Angles Vertical Angles Theorem

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Alternate Interior Angles Vertical Angles Theorem ASA Congruence Theorem

Given: seg QR  seg UT, RS  TS, QS=10, US=10 Prove: ΔQRS  ΔUTS

Proof Statements Reasons 1. 1. given 2. QS=US 2. subst. prop. = 3. Seg QS  seg US 3. Def of  segs. 4. Δ QRS  Δ UTS 4. SSS post seg QR  seg UT, RS  TS, QS=10, US=10

Given: seg WX  seg. XY, seg VX  seg ZX, Prove: Δ VXW  Δ ZXY 1 2 Y V

Proof Statements Reasons 1. seg WX  seg. XY 1. given seg. VX  seg ZX 2. 1  2 2. vert s thm 3. Δ VXW  Δ ZXY 3. SAS post

Given: seg RS  seg RQ and seg ST  seg QT Prove: Δ QRT  Δ SRT.

Proof Statements Reasons 1. Seg RS  seg RQ 1. Given seg ST  seg QT 2. Seg RT  seg RT 2. Reflex prop  3. Δ QRT  Δ SRT 3. SSS post

Given: seg DR  seg AG and seg AR  seg GR Prove: Δ DRA  Δ DRG.

Proof Statements seg DR  seg AG Seg AR  seg GR 2. seg DR  Seg DR 3.DRG & DRA are rt. s 4.DRG   DRA 5. Δ DRG  Δ DRA Reasons Given reflex. Prop of   lines form 4 rt. s 4. Rt. s thm 5. SAS post.

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