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Proving Triangle Congruence Using AAS, ASA, and Vertical Angles Theorem

This document explores the methods to prove triangle congruence, specifically focusing on scenarios where triangles can be shown to be congruent using the Angle-Angle-Side (AAS) and Angle-Side-Angle (ASA) theorems, along with the Vertical Angles Theorem. It outlines examples where pairs of corresponding angles and sides are congruent, and discusses situations where insufficient information may prevent proof of congruence. The importance of reasoning and logical deduction in the proof process is emphasized through detailed steps and proofs.

brady-adkins
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Proving Triangle Congruence Using AAS, ASA, and Vertical Angles Theorem

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  1. Developing a Triangle Proof

  2. Is it possible to prove the triangles are congruent? If so, state the theorem you would use. Explain your reasoning. 1. Developing Proof

  3. A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG. 1. Developing Proof

  4. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. 2. Developing Proof

  5. B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent. 2. Developing Proof

  6. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Given: UZ ║WX and UW║WX. 3. Developing Proof 1 2 3 4

  7. The two pairs of parallel sides can be used to show 1  3 and 2  4. Because the included side WZ is congruent to itself; ∆WUZ  ∆ZXW by the ASA Congruence. 3. Developing Proof 1 2 3 4

  8. Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC Use the fact that AD║EC to identify a pair of congruent angles. 4. Proving Triangles are Congruent

  9. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. 4. Proof:

  10. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Given 4. Proof:

  11. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given 4. Proof:

  12. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles 4. Proof:

  13. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Vertical Angles Theorem 4. Proof:

  14. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Vertical Angles Theorem ASA Congruence Theorem 4. Proof:

  15. Note: • You can often use more than one method to prove a statement. In Example 3, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.

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