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Objectives Use the SSS Postulate Use the SAS Postulate Use the HL Theorem Use ASA Postulate Use AAS Theorem CPCTC Theorem Vocabulary Bisect: to cut into two equal parts Postulate (SSS) Side-Side-Side Postulate 3 sides of one Δ are to 3 sides of another Δ, then the Δs are . If More on the SSS Postulate If seg AB seg ED, seg AC seg EF, & seg BC seg DF, then ΔABC ΔEDF. E A F C B D EXAMPLE 1 Use the SSS Congruence Postulate Write a proof. GIVEN PROVE Proof KL NL, KM KLM NM NLM It is given that KL NL and KM By the Reflexive Property, LM So, by the SSS Congruence Postulate, KLM NLM NM LN. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 1. DFG HJK SOLUTION Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. Side DG HK, Side DF JH,and Side FG JK. So by the SSS Congruence postulate, Yes. The statement is true. DFG HJK. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 2. ACB CAD SOLUTION GIVEN : BC PROVE : PROOF: AD ACB CAD It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD. GUIDED PRACTICE for Example 1 Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 3. QPT RST SOLUTION GIVEN : QT TR , PQ SR, PT TS PROVE : RST PROOF: QPT It is given that QT TR, PQ SR, PT TS. So by SSS congruence postulate, QPT RST. Yes the statement is true. Postulate (SAS) Side-Angle-Side Postulate 2 sides and the included of one Δ are to 2 sides and the included of another Δ, then the 2 Δs are . If More on the SAS Postulate seg BC seg YX, seg AC seg ZX, & C X, then ΔABC ΔZXY. B Y If ( A C X Z EXAMPLE 2 Use the SAS Congruence Postulate Write a proof. GIVEN BC DA, BC AD ABC PROVE CDA STATEMENTS S REASONS 1. BC DA 1. Given 2. BC AD 2. Given A 3. S 4. BCA AC DAC CA 3. Alternate Interior Angles Theorem 4. Reflexive Property of Congruence EXAMPLE 2 Use the SAS Congruence Postulate STATEMENTS 5. ABC CDA REASONS 5. SAS Congruence Postulate Example 3: Given: RS RQ and ST QT Prove: Δ QRT Δ SRT. S Q R T Q Example 3: R T Statements Reasons________ 1. RS RQ; ST QT 1. Given 2. RT RT 3. Δ QRT Δ SRT Postulate 2. Reflexive 3. SSS R Example 4: Given: DR AG and AR GR Prove: Δ DRA Δ DRG. D A R G Example 4: 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 A R G Theroem (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 . Note: Right Triangles Only Postulate (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. Theorem (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 nonincluded 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 A D B F Paragraph Proof C E 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. Theroem (CPCTC) Corresponding Parts of Congruent Triangles are Congruent When two triangles are congruent, there are 6 facts that are true about the triangles: the triangles have 3 sets of congruent (of equal length) sides and the triangles have 3 sets of congruent (of equal measure) angles. Use this after you have shown that two figures are congruent. Then you could say that Corresponding parts of the two congruent figures are also congruent to each other. 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: 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: 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. Writing Proofs Proofs are used to prove what you are finding. Geometric proofs can be written in one of two ways: two columns, or a paragraph. A paragraph proof is only a two-column proof written in sentences List the given statements and then list the conclusion to be proved Draw a figure and mark the figure accordingly along with your proofs Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given 2. Given 3. If || lines, then alt. int. s are 4. Vertical Angles Theorem 5. ASA Congruence Postulate