Other methods of Proving Triangles Congruent (AAS), (HL)

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Presentation transcript:

Other methods of Proving Triangles Congruent (AAS), (HL) 4-5

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

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

GUIDED PRACTICE for Examples 1 and 2 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. RTS UTV

GUIDED PRACTICE for Examples 1 and 2 Rewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof. GIVEN ABC PROVE 3 = 180° 1 m 2 + STATEMENTS REASONS 1. Draw BD parallel to AC . Parallel Postulate 2. Angle Addition Postulate and definition of straight angle 4 m 2 5 + = 180° 3. Alternate Interior Angles Theorem 1 4 , 3 5 4. Definition of congruent angles 1 m = 4 3 5 , 5. Substitution Property of Equality 1 m 2 3 + = 180°

EXAMPLE 3 Write a flow proof In the diagram, CE BD and ∠ CAB CAD. Write a flow proof to show ABE ADE GIVEN CE BD, ∠ CAB CAD PROVE ABE ADE

EXAMPLE 4 Standardized Test Practice

EXAMPLE 4 Standardized Test Practice The locations of tower A, tower B, and the fire form a triangle. The dispatcher knows the distance from tower A to tower B and the measures of A and B. So, the measures of two angles and an included side of the triangle are known. 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.

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

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

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

GUIDED PRACTICE for Examples 3 and 4 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.