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

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

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 Prove that SVR UVR STATEMENTS REASONS SV VU Given SVR RVU Definition of line RV VR Reflexive Property of Congruence SVR UVR SAS Congruence Postulate

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

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

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. Redraw ACB and DBC side by side with corresponding parts in the same position.

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

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

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.

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.

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.

Daily Homework 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 .