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Section 4.4 Use the SAS Congruence Postulate Posutlate 20 Side-Angle- Side (SAS) Congruence IF two sides and included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If: Side RS UV Angle R U Side RT UW RST UVW EXAMPLE 1 Use the SAS Congruence Postulate Write a proof. GIVEN BC DA, BC AD PROVE ABC CDA STATEMENTS REASONS S 1. BC DA 1. Given 2. BC AD 2. Given A 3. BCA DAC 3. Alternate Interior Angles Theorem S 4. AC CA 4. Reflexive Property of Congruence EXAMPLE 1 Use the SAS Congruence Postulate STATEMENTS REASONS 5. ABC CDA 5. SAS Congruence Postulate EXAMPLE 1 Use the SAS Congruence Postulate Given: MP NP, OP bisects MPN Prove: MOP NOP 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. ANSWER MRS and MPQ are congruent by the SAS Congruence Postulate. 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 2. SVR RVU 2. Definition of line 3. RV VR 3. Reflexive Property of Congruence 4. SVR UVR 4. SAS Congruence Postulate GUIDED PRACTICE for Examples 1 and 2 2. Prove that BSR DUT STATEMENTS REASONS 1. BS DU 1. Given 2. RBS TDU 2. Definition of line 3. RS UT 3. Given 4. BSR DUT 4. SAS Congruence Postulate EXAMPLE 2 Use SAS and properties of shapes In the diagram R is the center of the circle. If angle SRT is congruent to angle URT, what can you conclude about triangle SRT and Triangle URT? Section 4.4 Right Angles Right Triangles: In a right triangles the sides adjacent to the right angle are called the legs. The side opposite the right angle is called the hypotenuse of the right angle Section 4.4 Right Angles Theorem 4.5 Hypotenuse-Leg(HL) Congruence If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. ABC DEF EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem Write a proof. GIVEN WY XZ, WZ ZY, XY ZY PROVE WYZ XZY SOLUTION Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram. EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem STATEMENTS REASONS 1. WY XZ 1. Given 2. WZ ZY, XY ZY 2. Given 3. Z and Y are 3. Definition of lines right angles 4. WYZ and XZY are 4. Definition of a right right triangles. triangle L 5. ZY YZ 5. Reflexive Property of Congruence 6. WYZ XZY 6. HL Congruence Theorem EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem Given YW XZ , XY ZY Prove: XYW ZYW 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 You are given that PQ PS . By the Reflexive Property, RP RP . By the definition of perpendicular lines, both RPQ and RPS are right angles, so they are congruent. So, two sides and their included angle are congruent. ANSWER You can use the SAS Congruence Postulate to conclude that PQR PSR. EXAMPLE 4 Choose a postulate or theorem If you know that AB BC and ABD CBD, what postulate or theorem can you use conclude that ABD and CBD are congruent? 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 Use the diagram at the right. 4. Use the information in the diagram to prove that ACB DBC STATEMENTS REASONS 1. AC DB 1. Given 2. AB BC, CD BC 2. Given 3. C B 3. Definition of lines 4. ACB and DBC are 4. Definition of a right right triangles. triangle GUIDED PRACTICE for Examples 3 and 4 STATEMENTS REASONS L 5. BC CB 5. Reflexive Property of Congruence 6. ACB DBC 6. HL Congruence Theorem Summary Summary Summary: Summarize the major points How can you use two sides and an angle to prove triangles congruent? Day 1 p.243-246 1, 2 ,4-even, 9-18, 25-27 Day 2 p.243-246 19-24, 31-39, 42-48 even