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4.5 Prove Triangles Congruent by SAS and HL • You will use sides and angles to prove congruence. • Essential Question: How can you use two sides and an angle to prove triangles congruent? You will learn how to answer this question by using the SAS Post. and the HL Thm. Warm-Up Exercises Given: DF bisects CE, DC Prove: ∆CDF C ∆EDF DE DF bisects CE given CF EF def. of bisector F E D DC DE given DF DF Refl. Prop. of Segs ∆CDF SSS ∆ EDF Warm-Up1Exercises EXAMPLE Use the SAS Congruence Postulate Write a proof. GIVEN BC DA, BC AD ABC PROVE CDA STATEMENTS S REASONS 1. BC DA 1. Given 2. BC AD 2. Given A 3. S 4. BCA AC DAC CA 3. Alternate Interior Angles Theorem 4. Reflexive Property of Congruence Warm-Up1Exercises EXAMPLE Use the SAS Congruence Postulate STATEMENTS 5. ABC CDA REASONS 5. SAS Congruence Postulate Warm-Up2Exercises EXAMPLE 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. Is there only one way to match the ANSWER vertices to get a true congruence statement? MRS and MPQ are congruent by the SAS Congruence Postulate. Warm-Up Exercises 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 VU . of ABCD. Also, RT SU and SU 1. Prove that SVR UVR STATEMENTS REASONS 1. 1. Given 2. 3. 4. SV VU SVR RV RVU VR SVR UVR 2. Definition of line 3. Reflexive Property of Congruence 4. SAS Congruence Postulate Warm-Up Exercises GUIDED PRACTICE 2. Prove that for Examples 1 and 2 BSR DUT STATEMENTS REASONS 1. 1. Given 2. BS RBS 3. RS 4. DU BSR TDU 2. Definition of line 3. Given UT DUT 4. SAS Congruence Postulate Warm-Up3Exercises EXAMPLE Use the Hypotenuse-Leg Congruence Theorem Write a proof. GIVEN PROVE WY XZ, WZ ZY, XY ZY 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. Warm-Up3Exercises EXAMPLE Use the Hypotenuse-Leg Congruence Theorem STATEMENTS 1. WY 4. ZY 2. Given 3. Definition of Z and Y are lines right angles WYZ and XZY are 4. Definition of a right triangle right triangles. L 5. ZY 6. 1. Given XZ 2. WZ ZY, XY 3. REASONS WYZ YZ 5. Reflexive Property of Congruence XZY 6. HL Congruence Theorem Warm-Up4Exercises EXAMPLE 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? Warm-Up4Exercises EXAMPLE 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 PQR PSR. that Warm-Up Exercises 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. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 STATEMENTS REASONS L 5. BC 5. Reflexive Property of Congruence 6. ACB CB DBC 6. HL Congruence Theorem Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Use the diagram at the right. 4. Use the information in the diagram to ACB DBC prove that STATEMENTS 1. AC 2. AB REASONS 1. Given DB BC, CD B BC 2. Given 3. Definition of lines 3. C 4. ACB and DBC are 4. Definition of a right triangle right triangles. Daily Homework Quiz Warm-Up Exercises Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 1. ABE, CBD ANSWER SAS Post. Daily Homework Quiz Warm-Up Exercises Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 2. FGH, HJK ANSWER HL Thm. Daily Homework Quiz Warm-Up Exercises State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate. 3. ST YZ, RS XY ANSWER S Y. Daily Homework Quiz Warm-Up Exercises State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate. 4. T Z, RT XZ ANSWER ST YZ . • You will use sides and angles to prove congruence. • Triangles are congruent by the SAS Congruence Postulate. • Right triangles are congruent by the HL Congruence Theorem. • Essential Question: How can you use two sides and an angle to prove triangles congruent? You can prove triangles congruent if you know that two sides and the included angle of one triangle are congruent to two sides and the included angle of the other. If the triangles are right triangles, you can prove them congruent if they have congruent hypotenuses and a pair of congruent legs.