4.5 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Prove Triangles Congruent by ASA and AAS

4.5 Warm-Up ANSWER Yes; HL Thm. Tell whether the pair of triangles is congruent or not and why.

4.5 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. SOLUTION 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. a.

4.5 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. SOLUTION b. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent.

4.5 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. SOLUTION c. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate.

4.5 Example 2 Prove the Angle-Angle-Side Congruence Theorem. GIVEN: A D, C F, BC EF PROVE: ABC DEF

4.5 Guided Practice 1. In the diagram at the right, what postulate or theorem can you use to prove that RST VUT ? Explain. ANSWER AAS ;  RTS and  VTU are also congruent because they are also vertical angles.

4.5 Guided Practice 2. Rewrite the proof of the Triangle Sum Theorem on page 227 as a flow proof. Compare the flow proof with the two- column proof on page 227. PROVE: 3 = 180° 1m2mm++ GIVEN: ABC ANSWER

4.5 Guided Practice The two-column proof uses numbered steps with statements in the left column and a reason for each step in the right column. The flow proof shows the 5 main steps as boxes that are connected by arrows to show the flow of the logical argument.

4.5 Example 3 In the diagram, CE BD and  CAB CAD. Write a flow proof to show ABE ADE. GIVEN: CE BD,  CAB CAD PROVE: ABE ADE

4.5 Example 4 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.

4.5 Example 4 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. The correct answer is B. ANSWER

4.5 Lesson Quiz In Example 3, suppose ABE ADE is also given. What theorem or postulate besides ASA can you use to prove that 3. ABE ADE ? ANSWERAAS Congruence Theorem

4.5 Guided Practice 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. ANSWER No; no triangle is formed by the location of the fire and towers, so the fire could be anywhere between towers B and C.

4.5 Lesson Quiz Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. ANSWER ASA 1.

4.5 Lesson Quiz Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. ANSWER not necessarily congruent 2.

4.5 Lesson Quiz Write flow proof. Given : BD bisects ABC, A C Prove : ABD CBD 3. ANSWER