1. You will use sides and angles to prove congruence. Essential Question: How can you use two sides and an angle to prove triangles congruent? 4.5 Prove Triangles Congruent by SAS and HL You will learn how to answer this question by using the SAS Post. and the HL Thm.

2. Warm-Up Exercises Prove: ∆CDF ∆EDF Given: DF bisects CE, DC DE C F E D DF bisects CE DC DE CF EF def. of bisector given DF Refl. Prop. of Segs ∆CDF ∆ EDF SSS

3. Warm-Up Exercises 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

4. Warm-Up Exercises EXAMPLE 1 Use the SAS Congruence Postulate STATEMENTS REASONS 5. ABC CDA SAS Congruence Postulate 5.

5. Warm-Up Exercises 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 Is there only one way to match the vertices to get a true congruence statement?

6. 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 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

7. Warm-Up Exercises 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

8. Warm-Up Exercises 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

9. Warm-Up Exercises STATEMENTS REASONS EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem 1. WY XZ 1. 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 2. WZ ZY, XY ZY Given

10. Warm-Up Exercises 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 ?

11. Warm-Up Exercises 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

12. 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.

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

14. Warm-Up Exercises 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. AC DB 1. Given 2. AB BC, CD BC Given 4. Definition of a right triangle ACB and DBC are right triangles. 3. Definition of lines C B

15. Warm-Up Exercises Daily Homework Quiz Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 1. ABE, CBD ANSWER SAS Post.

16. Warm-Up Exercises Daily Homework Quiz Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 2. FGH, HJK ANSWER HL Thm.

17. Warm-Up Exercises Daily Homework Quiz 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.

18. Warm-Up Exercises Daily Homework Quiz State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate. ANSWER ST YZ. 4. T Z, RT XZ

19. You will use sides and angles to prove congruence. Essential Question: How can you use two sides and an angle to prove triangles congruent? Triangles are congruent by the SAS Congruence Postulate. Right triangles are congruent by the HL Congruence Theorem. 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.