2SSS Side-Side-Side Postulate If 3 sides of one Δ are to 3 sides of another Δ, then the Δs are .
3More on the SSS Postulate If AB ED, AC EF, & BC DF, then ΔABC ΔEDF.EDFABC
4EXAMPLE 1:Use the SSS Congruence PostulateWrite a proof.GIVENKL NL, KM NMPROVEKLM NLMProofIt is given thatKL NL and KM NMBy the Reflexive Property,LM LN.So, by the SSS Congruence Postulate,KLM NLM
5GUIDED PRACTICEYOUR TURN:Decide whether the congruence statement is true. Explain your reasoning.1.ACB CADSOLUTIONBC ADGIVEN :PROVE :ACB CADPROOF:It is given that BC AD By Reflexive propertyAC AC, But AB is not congruent CD.
6GUIDED PRACTICEYOUR TURN (continued):Therefore the given statement is false and ABC is notCongruent to CAD because corresponding sidesare not congruent
7GUIDED PRACTICEYOUR TURN:Decide whether the congruence statement is true. Explain your reasoning.QPT RST2.SOLUTIONQT TR , PQ SR, PT TSGIVEN :PROVE :QPT RSTPROOF:It is given that QT TR, PQ SR, PT TS. So bySSS congruence postulate, QPT RST. Yes, the statement is true.
8SAS 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 .
9More on the SAS Postulate If BC YX, AC ZX, & C X, then ΔABC ΔZXY.BY)(ACXZ
10EXAMPLE 2Example 2:Use the SAS Congruence PostulateWrite a proof.GIVENBC DA, BC ADPROVEABC CDASTATEMENTSREASONSGivenBC DASGivenBC ADBCA DACAlternate Interior Angles TheoremAAC CAReflexive Property of CongruenceS
12Given: DR AG and AR GR Prove: Δ DRA Δ DRG. Example 4:Given: DR AG and AR GR Prove: Δ DRA Δ DRG.DRAG
13Example 4 (continued):Statements_______ 1. DR AG; AR GR 2. DR DR 3.DRG & DRA are rt. s 4.DRG DRA 5. Δ DRG Δ DRAReasons____________1. Given2. Reflexive Property3. lines form 4 rt. s4. Right s Theorem5. SAS PostulateDRGA
14HL 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 .
15ASA 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.
16AAS 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.
17Proof of the Angle-Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EFProve: ∆ABC ∆DEFDABFCParagraph ProofYou 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
18Example 5:Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
19Example 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.
20Example 6:Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
21Example 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.
22Example 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.
23If || lines, then alt. int. s are Vertical Angles Theorem Example 7 (continued):Statements:BD BCAD ║ ECD CABD EBC∆ABD ∆EBCReasons:GivenIf || lines, then alt. int. s are Vertical Angles TheoremASA Congruence Postulate