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4-2 Triangle Congruence using SSS and SAS Postulate 4-1 Side-Side-Side (SSS) Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. A D DFG ABC C B G F Postulate 4-2 Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. D A C B 1. In DFG ABC G F ABP , which sides include B ? A P 2. In YXP which angle is between PX and XY ? Y B X 3. Which triangles can you prove congruent? Tell whether you would use SSS or ASA Postulate. D 4. What other information do you need to prove DWO DWG ? O 5. Can you prove SED BUT from the information given? Explain. 1 W G Triangle Congruence: ASA and AAS Theorem Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. D A B ____ G C F ______ by _____ Ex 1: Given M 4 30 P 0 4 N O 60 Q 0 R Write a congruence statement for the triangles above. Ex 2: Which side is included between R and F in FTR? Ex 3: Which angles in STU include US ? Theorem Angle-Angle-Side (AAS) Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. D A G C B DFG F Ex 4: Given: D A 5 420 B AB C 0 38 0 C F 5 100 420 G Write a congruence Statement for the triangles above. Ex 5: Can you prove ASA or AAS? If yes, state the triangle congruence and the postulate or theorem you used. Given: that MNOP is a parallelogram. P Y L A 2 Section 4-6: Right Triangle Congruence Use the following theorems for RIGHT TRIANGLES Only! Since we already know that we have a congruent angle, we only have to prove two other things are congruent. Fill in the name for the parts of the right triangle below. Congruence Patterns: LL, HA, LA and HL Leg – Leg (LL): When both sets of legs are congruent, we have congruent triangles. Hypotenuse – Angle (HA): When the hypotenuse of both triangles and another angle outside of the right angle are congruent, we have congruent triangles. Leg – Angle (LA): To prove LA, we need one leg and one other angle. OR Hypotenuse – Leg (HL): When the hypotenuse of both triangles and on the two legs is congruent, we can prove congruence using HL. 3 Examples: Using only the right triangle theorems, LL, HA, HL and LA, prove that the following triangles are congruent. 1. 2. 3. 4. FG HK , F G 5. LJK is isosceles with base LK , GJL IJK GL GI , KI GI 6. Q is the midpoint of PX. F H K G J I G P K L 4 Q Z X Sample Proofs for 4-2 (SSS and SAS) Given: Prove: AB DE , AB DE , BC FE B ∆ABF ∆DEC D C F A E B Given: Prove: ABC is isosceles with base AC D is the midpoint of AC ABD CBD A 5 D C Sample Proofs Using ASA and AAS 1. Given: B BD bisects ADC BD bisects ABC Prove: ABD CBD A C D 2. Given: D is the midpoint of MT B Z MB DZ B Z Prove: MDB DTZ M 6 D T Sample right triangle proofs: 1. Given: BX DY D Y A BA AY DC XC AY XC Prove: ABY CDX B C X D 2. Given: NT DA DA bisects NT ND TA Prove: SND STA N S A 7 T Examples for CPCTC – Corresponding Parts of Congruent Triangles are Congruent When you prove two triangles congruent, you use only three pairs of parts. After the two triangles have been shown to be congruent, the remaining three pairs of parts are congruent by CPCTC. You will need to use CPCTC if you are to prove something other than congruent triangles. Example 1 Given: C is the midpoint of AE and BD A Prove: AB ED and AB ED B C D E 8 Example 2 Given: ∆UJY is isosceles triangle with vertex J JR bisects UJY J Prove: R is the midpoint of UY and JR UY U 9 R Y 10 11