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Proving Triangles Congruent Side-Side-Side Postulate Side-Angle-Side Postulate Angle-Side-Angle Postulate Angle-Angle-Side Theorem • How much do you need to know to prove that triangles are congruent? • We know that if we can show all three sides and all three angles are congruent, we can show that the triangles are congruent. Postulate Side-Side-Side Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. M P If Side MN ≅ QR Side NP ≅ RS Q N Side PM ≅ SQ Then ∆MPN ≅ ∆QRS by SSS R S Postulate Side-Angle-Side Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. X If Side PQ ≅ WX Q Angle ∠ Q ≅ ∠X Side QS ≅ XY W P then ∆PQS ≅ ∆WXY by SAS S Y Postulate Angle-Side Angle (ASA) 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 two triangles are congruent. If Angle ∠ A ≅ ∠D B Side AB ≅ DE Angle ∠B ≅ ∠E then ∆ABC ≅ ∆DEF by ASA A C E D F EXAMPLE 1 Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem. a. C E H b. I D K G J F NO, there is no AAA theorem or postulate of congruence. YES, HJ ≅ HJ by the reflexive property, so the triangles are congruent by ASA Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle, then the two triangles are congruent. If Angle ∠A ≅ ∠D, Angle ∠C ≅ ∠F, and B Side AB≅ DE, Then ∆ABC ≅ ∆DEF by AAS A C E D F Example 2 Determine if there is enough information to determine if the pairs of triangles are congruent. If there is enough information, indicate the postulate or theorem that can be used. Yes, SAS a) b) Determine if there is enough information to determine if the pairs of triangles are congruent. If there is enough information, indicate the postulate or theorem that can be used. Yes, ASA c) Determine if there is enough information to determine if the pairs of triangles are congruent. If there is enough information, indicate the postulate or theorem that can be used. No Name the included side between: 1. XZ ∠X and ∠Z _____ T Z S R X Y F A C B D E J L M K N O NO 2. ∠N and ∠O _______ T Z S R X Y F A C B D E J L M K N O 3. ∠A and ∠B ________ AB T Z S R X Y F A C B D E J L M K N O Name the include angle between: 4. TR and ST ________ ∠T T Z S R X Y F A C B D E J L M K N O ∠K 5. KL and KJ ________ T Z S R X Y F A C B D E J L M K N O E ________ 6. FE and DE ∠ T Z S R X Y F A C B D E J L M K N O Tests for Congruent Triangles SSS __________ postulate ASA __________ postulate SAS __________ postulate AAS __________ theorem Name the postulate or theorem that you can use to prove the triangles congruent and write a congruence statement. 7 U 8 V S Y W X X T T W ASA ∆TUV ≅ ∆WXY U V SSS ∆STU ≅ ∆VXW 9 C H G 10 AAS J H D E K P F SAS ∆CDE ≅ ∆FGH ∆JHK ≅ ∆MLP M L Mark the third congruence that must be given to prove ∆ABC ≅ ∆DEF using the indicated postulate or theorem. 11. ASA Postulate F D B E A C A 12. AAS Theorem B C D E F D B 13. SSS Postulate C F A E E 14. SAS Postulate D A F B C Determine if enough information is given to prove the triangles congruent. If there is, state the postulate or the theorem and write a congruence statement. 15. A E ASA ∆ACB ≅ ∆DCE C B 16. D Y Z SSS ∆XYZ ≅ ∆ZWX X W G 17. NO K 19. H 18. J U K H N AAS ∆JKU ≅ ∆LKU L L SAS ∆JKH ≅ ∆KJLJ 20. AAS T N ∆TSN ≅ ∆USH K L H S U 21. 22. C M H SAS ∆ABC ≅ ∆DCA Q U SAS ∆MQU ≅ ∆CUQ HOMEWORK: Chapter 13 Review