Warm Up Lesson Presentation Lesson Quiz Prove Triangles Congruent by SAS and HL Warm Up Lesson Presentation Lesson Quiz

Warm-Up Prove: ∆CDF ∆EDF Given: DF bisects CE, DC DE C F E D ANSWER It is given that DC DE and DF bisects CE. CF EF by the def. of bisector. DF DF by the Refl. Prop. of Segs. So ∆CDF ∆ EDF by the SSS Post.

Example 1 Write a proof. GIVEN: BC DA, BC AD PROVE: ABC CDA STATEMENTS REASONS BC DA S Given BC AD Given BCA DAC A Alternate Interior Angles Theorem AC CA S Reflexive Property of Congruence 5. ABC CDA 5. SAS Congruence Postulate

Example 2 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 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 Prove that SVR UVR ANSWER

Guided Practice 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 Prove that BSR DUT ANSWER

Guided Practice

Example 3 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 STATEMENTS REASONS WY XZ H Given WZ ZY, XY ZY Given Z and Y are right angles Definition of lines WYZ and XZY are right triangles. Definition of a right triangle L ZY YZ Reflexive Property of Congruence WYZ XZY HL Congruence Theorem

Example 4 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 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 Use the diagram at the right. Redraw ACB and DBC side by side with corresponding parts in the same position. ANSWER

Guided Practice Use the diagram at the right. Use the information in the diagram to prove that ACB DBC ANSWER

Lesson 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.

Lesson 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.

Lesson 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.

Lesson Quiz 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 .