EXAMPLE 1 Use the SAS Congruence Postulate Write a proof. GIVEN PROVE STATEMENTS REASONS BC DA, BC AD ABC CDA 1. Given 1. BC DA S Given 2. BC AD 3. BCA DAC 3. Alternate Interior Angles Theorem A 4. AC CA Reflexive Property of Congruence S

EXAMPLE 1 Use the SAS Congruence Postulate STATEMENTS REASONS 5. ABC CDA SAS Congruence Postulate 5.

EXAMPLE 2 Use SAS and properties of shapes In the diagram, QS and RP pass through the center M of the circle. What can you conclude about MRS and MPQ ? SOLUTION Because they are vertical angles, PMQ RMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MS are all equal. MRS and MPQ are congruent by the SAS Congruence Postulate. ANSWER

GUIDED PRACTICE for Examples 1 and 2 In the diagram, ABCD is a square with four congruent sides and four right angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT SU and. SU VU 1. Prove that SVR UVR STATEMENTS REASONS 1. SV VU 1. Given 3. RV VR Reflexive Property of Congruence 2. SVR RVU Definition of line 4. SVR UVR SAS Congruence Postulate

GUIDED PRACTICE for Examples 1 and 2 2. Prove that BSR DUT STATEMENTS REASONS 1. Given BS DU 2. RBS TDU Definition of line 3. RS UT Given 4. BSR DUT SAS Congruence Postulate

EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem Write a proof. SOLUTION Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram. GIVEN WY XZ, WZ ZY, XY ZY PROVE WYZ XZY

STATEMENTS REASONS EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem 1. WY XZ 1. Given H 2. WZ ZY, XY ZY Given 4. Definition of a right triangle WYZ and XZY are right triangles. L ZY YZ 5. Reflexive Property of Congruence 6. WYZ XZY 6. HL Congruence Theorem 3. Definition of lines Z and Y are right angles

EXAMPLE 4 Choose a postulate or theorem Sign Making You are making a canvas sign to hang on the triangular wall over the door to the barn shown in the picture. You think you can use two identical triangular sheets of canvas. You know that RP QS and PQ PS. What postulate or theorem can you use to conclude that PQR PSR?

EXAMPLE 4 Choose a postulate or theorem SOLUTION RPQ and RPS are right angles, so they are congruent. So, two sides and their included angle are congruent. You are given that PQ PS. By the Reflexive Property, RP RP. By the definition of perpendicular lines, both You can use the SAS Congruence Postulate to conclude that. PQR PSR ANSWER

GUIDED PRACTICE for Examples 3 and 4 Use the diagram at the right. 3. Redraw ACB and DBC side by side with corresponding parts in the same position.

GUIDED PRACTICE for Examples 3 and 4 4. Use the diagram at the right. Use the information in the diagram to prove that ACB DBC STATEMENTS REASONS 1. AB DC 1. Given H 2. AC BC, DB BC Given 4. Definition of a right triangle ACB and DBC are right triangles. 3. Definition of lines C B

GUIDED PRACTICE for Examples 3 and 4 STATEMENTS REASONS L BC CB 5. Reflexive Property of Congruence 6. ACB DBC 6. HL Congruence Theorem