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Transcript

4.5 Proving Triangles are Congruent: ASA and AAS Geometry Objectives: 1. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem 2. Use congruence postulates and theorems in real-life problems. Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the B included side of one triangle are congruent to two angles and the C included side of a second triangle, then the triangles are congruent. A E F D Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem • If two angles and a B non-included side of one triangle are congruent to two angles and the corresponding non- C included side of a second triangle, then the triangles are congruent. A E F D Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF Prove: ∆ABC ∆DEF B A E C F D Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF C is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF. B A E F D Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. H E G F J Ex. 1 Developing Proof 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. H E G F J Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. N M Q P Ex. 1 Developing Proof 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. N M Q P Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW ║WX. U 1 2 W 3 4 X Z Ex. 1 Developing Proof 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 Postulate. U 1 2 W 3 4 X Z Ex. 2 Proving Triangles are Congruent 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. C A B D E C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem C A Proof: B D Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem 5. ASA Congruence Theorem Note: • You can often use more than one method to prove a statement. In Example 2, 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.