# Proving Δs are  : SSS, SAS, HL, ASA, & AAS

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Proving Δs are  : SSS, SAS, HL, ASA, & AAS

SSS Side-Side-Side  Postulate
If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

More on the SSS Postulate
If AB  ED, AC  EF, & BC  DF, then ΔABC  ΔEDF. E D F A B C

EXAMPLE 1: Use the SSS Congruence Postulate Write a proof. GIVEN KL NL, KM NM PROVE KLM NLM Proof It is given that KL NL and KM NM By the Reflexive Property, LM LN. So, by the SSS Congruence Postulate, KLM NLM

GUIDED PRACTICE YOUR TURN: Decide whether the congruence statement is true. Explain your reasoning. 1. ACB CAD SOLUTION BC AD GIVEN : PROVE : ACB CAD PROOF: It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD.

GUIDED PRACTICE YOUR TURN (continued): Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent

GUIDED PRACTICE YOUR TURN: Decide whether the congruence statement is true. Explain your reasoning. QPT RST 2. SOLUTION QT TR , PQ SR, PT TS GIVEN : PROVE : QPT RST PROOF: It is given that QT TR, PQ SR, PT TS. So by SSS congruence postulate, QPT RST. Yes, the statement is true.

SAS Side-Angle-Side  Postulate
If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

More on the SAS Postulate
If BC  YX, AC  ZX, & C  X, then ΔABC  ΔZXY. B Y ) ( A C X Z

EXAMPLE 2 Example 2: Use the SAS Congruence Postulate Write a proof. GIVEN BC DA, BC AD PROVE ABC CDA STATEMENTS REASONS Given BC DA S Given BC AD BCA DAC Alternate Interior Angles Theorem A AC CA Reflexive Property of Congruence S

EXAMPLE 2 Example 2 (continued): STATEMENTS REASONS ABC CDA SAS Congruence Postulate

Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG.
Example 4: Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG. D R A G

Example 4 (continued): Statements_______ 1. DR  AG; AR  GR 2. DR  DR 3.DRG & DRA are rt. s 4.DRG   DRA 5. Δ DRG  Δ DRA Reasons____________ 1. Given 2. Reflexive Property 3.  lines form 4 rt. s 4. Right s Theorem 5. SAS Postulate D R G A

HL Hypotenuse - Leg  Theorem
If the hypotenuse and a leg of a right Δ are  to the hypotenuse and a leg of a second Δ, then the 2 Δs are .

ASA Angle-Side-Angle 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.

AAS Angle-Angle-Side 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.

Proof of the Angle-Angle-Side (AAS) Congruence Theorem
Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF D A B F C Paragraph Proof You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. E

Example 5: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 5 (continued): In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

Example 6: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 6 (continued): In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

Example 7: 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.

If || lines, then alt. int. s are  Vertical Angles Theorem
Example 7 (continued): Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given If || lines, then alt. int. s are  Vertical Angles Theorem ASA Congruence Postulate

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