Presentation on theme: "4.3 to 4.5 Proving Δs are : SSS, SAS, HL, ASA, & AAS"— Presentation transcript:
1 4.3 to 4.5 Proving Δs are : SSS, SAS, HL, ASA, & AAS
2 Objectives Use the SSS Postulate Use the SAS Postulate Use the HL TheoremUse ASA PostulateUse AAS Theorem
3 Postulate 19 (SSS) Side-Side-Side Postulate If 3 sides of one Δ are to 3 sides of another Δ, then the Δs are .
4 More on the SSS Postulate If seg AB seg ED, seg AC seg EF, & seg BC seg DF, then ΔABC ΔEDF.EDFABC
5 EXAMPLE 1Use 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
6 Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.GUIDED PRACTICEfor Example 1Decide whether the congruence statement is true. Explain your reasoning.SOLUTIONYes. The statement is true.DFG HJKSide DG HK, Side DF JH,and Side FG JK.So by the SSS Congruence postulate, DFG HJK.
7 GUIDED PRACTICEfor Example 1Decide whether the congruence statement is true. Explain your reasoning.2.ACB CADSOLUTIONBC ADGIVEN :PROVE :ACB CADPROOF:It is given that BC AD By Reflexive propertyAC AC, But AB is not congruent CD.
8 GUIDED PRACTICEfor Example 1Therefore the given statement is false and ABC is notCongruent to CAD because corresponding sidesare not congruent
9 GUIDED PRACTICEfor Example 1Decide whether the congruence statement is true. Explain your reasoning.QPT RST3.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.
10 Postulate 20 (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 .
11 More on the SAS Postulate If seg BC seg YX, seg AC seg ZX, & C X, then ΔABC ΔZXY.BY)(ACXZ
12 EXAMPLE 2Use the SAS Congruence PostulateWrite a proof.GIVENBC DA, BC ADPROVEABC CDASTATEMENTSREASONSGivenBC DASGivenBC ADBCA DACAlternate Interior Angles TheoremAAC CAReflexive Property of CongruenceS
13 EXAMPLE 2Use the SAS Congruence PostulateSTATEMENTSREASONSABC CDASAS Congruence Postulate
14 Given: RS RQ and ST QT Prove: Δ QRT Δ SRT. Example 3:Given: RS RQ and ST QT Prove: Δ QRT Δ SRT.SQRT
16 Given: DR AG and AR GR Prove: Δ DRA Δ DRG. Example 4:Given: DR AG and AR GR Prove: Δ DRA Δ DRG.DRAG
17 Example 4: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
18 Theroem 4.5 (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 .
19 Postulate 21(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.
20 Theorem 4.6 (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.
21 Proof 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
22 Example 5:Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
23 Example 5: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.
24 Example 6:Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
25 Example 6: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.
26 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.
27 Proof: Statements: BD BC AD ║ EC D C ABD EBC ∆ABD ∆EBC Reasons:GivenIf || lines, then alt. int. s are Vertical Angles TheoremASA Congruence Postulate