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4.34.4 Proving Triangles Congruent Using SSS, SAS, ASA, AAS (work).notebook November 09, 2015 4.3 ‐ 4.4 PROVING TRIANGLES CONGRUENT Postulate 19: SSS Postulate If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. C X M H A Z Example 1 Given STU with vertices S(0, 5), T(0, 0), and U(‐2, 0) and XYZ with vertices X(4, 8), Y(4, 3), and Z(6, 3), determine if ~ XYZ. STU = 4.34.4 Proving Triangles Congruent Using SSS, SAS, ASA, AAS (work).notebook November 09, 2015 Postulate 20: 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 triangles are congruent. B The included angle is the angle formed by two given sides. W J U K G Example 2 Write a proof for the following. Given: X is the midpoint of BD. X is the midpoint of AC. ~ BXA Prove: DXC = Statements A D X C Reasons B 4.34.4 Proving Triangles Congruent Using SSS, SAS, ASA, AAS (work).notebook November 09, 2015 Postulate 21: ASA Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles L are congruent. The included side is the side of the triangle formed between two given angles. R Y S Q D V Example 3 Write a proof for the following. ~ US Given: VR RS, UT SU, RS = ~ TUS Prove: VRS = Statements Reasons R S U T 4.34.4 Proving Triangles Congruent Using SSS, SAS, ASA, AAS (work).notebook November 09, 2015 Theorem 4‐6: AAS Theorem If two angles and a nonincluded side of a triangle are congruent to the corresponding two angles and a side of a second triangle, the two triangles are congruent. V C T N O A Example 4 Write a two‐column proof. Given: PSU ~ = PTR SU ~ = TR ~ TRP Prove: SUP = T S R U P Statements Reasons 4.34.4 Proving Triangles Congruent Using SSS, SAS, ASA, AAS (work).notebook November 09, 2015 Example 5 Write a two‐column proof. Given: 1 and 2 are right angles. S ST ~ = TP 5 ~ PTR Prove: STR = 1 T 3 4 2 R 6 P Statements Reasons Example 6 Write a two‐column proof. A Given: BE bisects AD. ~ D A= ~ DEC Prove: ABE = B Statements Reasons C 1 E 2 D 4.34.4 Proving Triangles Congruent Using SSS, SAS, ASA, AAS (work).notebook November 09, 2015 Theorem 4‐8: HL Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. F B M Z H Example 7 Write a two‐column proof. Given: AE CD AC ~ =AD Prove: ACE ~ = ADE Statements A A C Reasons 1 2 E D