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Proving Triangle Congruence by ASA and AAS Theorems

This guide explores the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) congruence postulates. According to the ASA postulate, two triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding angles and side of another triangle. Similarly, the AAS theorem states that if two angles and a non-included side of one triangle are congruent to the two angles and the corresponding side of another triangle, then the triangles are congruent. This guide includes examples and proofs for better understanding.

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Proving Triangle Congruence by ASA and AAS Theorems

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  1. 4.10 Prove Triangles Cong. by ASA and AAS E B F D C A 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.

  2. 4.10 Prove Triangles Cong. by ASA and AAS E B F D C A 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.

  3. 4.10 Prove Triangles Cong. by ASA and AAS t p q Theorem 4.14: Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pair of alternate interior angles are _________. 4 congruent 5

  4. 4.10 Prove Triangles Cong. by ASA and AAS Identify congruent triangles Example 1 Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. a. • There is not enough information to prove the triangles congruent, because no _______ are known to be the congruent. sides

  5. 4.10 Prove Triangles Cong. by ASA and AAS Identify congruent triangles Example 1 Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. b. non-included • Two pairs of angles and a ______________ pair of sides are congruent. The triangles are congruent by the _____________________________. AAS Congruence Theorem

  6. 4.10 Prove Triangles Cong. by ASA and AAS Identify congruent triangles Example 1 Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. c. • The vertical angles are congruent, so two pairs of angles and their _______________ are congruent. The triangles are congruent by the ____________________________. included sides ASA Congruence Postulate

  7. 4.10 Prove Triangles Cong. by ASA and AAS • 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 U T W Checkpoint. Complete the following exercises. you have two pairs of angles and a non-included pair of sides. The triangles are congruent by the AAS Congruence Theorem

  8. 4.10 Prove Triangles Cong. by ASA and AAS In the diagram, 1 4 and CF bisects ACE. Write a flow proof to show CBF CDF. C D B 2 3 1 4 E A F Def. of _______________ angles Cong. Supps. Thm. _________________________ Write a flow proof Example 2 Given: Prove: supplements supplements Given supplementary Given Reflexive Prop. AAS Congruence Theorem

  9. 4.10 Prove Triangles Cong. by ASA and AAS C • In the diagram, CF bisects ACE and BFD. Write a flow proof to show CBF CDF. D B 2 3 1 4 E A F Def. of bisector Def. of bisector Checkpoint. Complete the following exercises. Given: Prove: Given Given Reflexive Prop. ASA Congruence Postulate

  10. 4.10 Prove Triangles Cong. by ASA and AAS Pg. 267, 4.10 #1-20

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