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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.

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EXAMPLE 1

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  1. Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. The triangles are congruent by the AAS Congruence Theorem. EXAMPLE 1 Identify congruent triangles SOLUTION

  2. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate. EXAMPLE 1 Identify congruent triangles

  3. C F, BC EF A D, GIVEN ABCDEF PROVE EXAMPLE 2 Prove the AAS Congruence Theorem Prove the Angle-Angle-Side Congruence Theorem. Write a proof.

  4. In the diagram at the right, what postulate or theorem can you use to prove that ? Explain. RSTVUT STATEMENTS REASONS Given S U Given RS UV The vertical angles are congruent RTSUTV for Examples 1 and 2 GUIDED PRACTICE SOLUTION

  5. RTSUTV for Examples 1 and 2 GUIDED PRACTICE ANSWER Therefore are congruent because vertical angles are congruent so two pairs of angles and a pair of non included side are congruent. The triangle are congruent by AAS Congruence Theorem.

  6. ABC GIVEN m 1 + m 2 + m 3 = 180° PROVE STATEMENTS REASONS Rewrite the proof of the Triangle Sum Theorem on page219as a flow proof. 1. Parallel Postulate 1. Draw BDparallel to AC. 2. m 4 + m 2 + m 5 2. = 180° Angle Addition Postulate and definition of straight angle , 4 3. 1 5 3. 3 Alternate Interior Angles Theorem m 4 m 5 4. m 1 = , m 3 = 4. Definition of congruent angles 5. m 1 + m 2 + m 3 5. Substitution Property of Equality = 180° for Examples 1 and 2 GUIDED PRACTICE

  7. GIVEN CE BD, ∠ CAB CAD ABEADE PROVE EXAMPLE 3 Write a flow proof In the diagram,CE BDand∠ CAB CAD. ABEADE Write a flow proof to show

  8. EXAMPLE 4 Standardized Test Practice

  9. The locations of tower A, tower B, and the fire form a triangle. The dispatcher knows the distance from tower A to tower Band the measures of Aand B. So, the measures of two angles and an included side of the triangle are known. EXAMPLE 4 Standardized Test Practice By the ASA Congruence Postulate, all triangles with these measures are congruent. So, the triangle formed is unique and the fire location is given by the third vertex. Two lookouts are needed to locate the fire.

  10. EXAMPLE 4 Standardized Test Practice ANSWER The correct answer is B.

  11. STATEMENTS REASONS ABE ADE Given AEB AED Definition of right triangle Both are right angle triangle. ABE ADE Reflexive Property of Congruence AAS Congruence Theorem BD DB for Examples 3 and 4 GUIDED PRACTICE In Example 3, suppose ABEADEis also given. What theorem or postulate besides ASA can you use to prove that ABEADE? SOLUTION

  12. What If?In Example 4, suppose a fire occurs directly between tower Band tower C. Could towers Band Cbe used to locate the fire? Explain The locations of tower B, tower C, and the fire form a triangle. The dispatcher knows the distance from tower B to tower Cand the measures of Band C. So, he knows the measures of two angles and an included side of the triangle. for Examples 3 and 4 GUIDED PRACTICE SOLUTION Proved by ASA congruence

  13. for Examples 3 and 4 GUIDED PRACTICE By the ASA Congruence Postulate, all triangles with these measures are congruent. No triangle is formed by the location of the fire and tower, so the fire could be anywhere between tower B and C.

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