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4.10 Prove Triangles Cong. by ASA and AAS 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 E B congruent. A C D F If Angle A _____, D Side AC _____, DF and Angle C _____ F then ABC ______ DEF 4.10 Prove Triangles Cong. by ASA and AAS Theorem 4.13: Angle-Angle-Side (AAS) 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 two triangles are congruent. E B A C D F If Angle A _____, F and D Angle C _____, then ABC ______ DEF Side BC _____, EF 4.10 Prove Triangles Cong. by ASA and AAS Theorem 4.14: Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pair of alternate interior angles are congruent _________. t p 4 5 q 4 5 4.10 Prove Triangles Cong. by ASA and AAS Example 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. a. a. There is not enough information to prove the triangles congruent, because no _______ sides are known to be the congruent. 4.10 Prove Triangles Cong. by ASA and AAS Example 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. b. b. Two pairs of angles and a ______________ non-included pair of sides are congruent. The triangles are congruent by the _____________________________. AAS Congruence Theorem 4.10 Prove Triangles Cong. by ASA and AAS Example 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. c. c. The vertical angles are congruent, so two pairs of angles and their included sides are congruent. _______________ The triangles are congruent by the ____________________________. ASA Congruence Postulate 4.10 Prove Triangles Cong. by ASA and AAS Checkpoint. Complete the following exercises. 1. Can STW and VWT be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. S V 60 o U 60 o T W Since TW WT, you have two pairs of angles and a non-included pair of sides. The triangles are congruent by the AAS Congruence Theorem 4.10 Prove Triangles Cong. by ASA and AAS C Example 2 Write a flow proof In the diagram, 1 4 and CF bisects ACE. Write a flow proof to show CBF CDF. A Given: 1 4, CF bisects ACE. Prove: ΔCBF ΔCDF 1 4, Given 3 D 4 B 2 1 E F supplements 1 and 2 are _________ supplements 3 and 4 are _________ CF bisects ACE. supplementary angles Def. of _______________ Given 2 ___ 3 CF CF Cong. Supps. Thm. Reflexive Prop. ΔCBF ΔCDF ECF ACF _____ Def. of bisector AAS Congruence Theorem _________________________ 4.10 Prove Triangles Cong. by ASA and AAS Checkpoint. Complete the following exercises. C 2. In the diagram, CF bisects ACE and BFD. Write a flow proof to B 2 1 show CBF CDF. A 3 D 4 F Given: CF bisects ACE, CF bisects BFD. Prove: ΔCBF ΔCDF CF bisects BFD CF bisects ACE Given Given ACF ECF Def. of bisector CF CF Reflexive Prop. BFC DFC Def. of bisector ΔCBF ΔCDF ASA Congruence Postulate E 4.10 Prove Triangles Cong. by ASA and AAS Pg. 267, 4.10 #1-20