4.4 Prove triangles are congruent by SAS and HL

4.4 Prove triangles are congruent by SAS and HL Objective: You will use sides and angles to prove congruence.

Side-Angle-Side (SAS) Congruence Postulate • If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

BC DA,BC AD ABCCDA STATEMENTS REASONS S BC DA Given Given BC AD BCADAC A Alternate Interior Angles Theorem S ACCA Reflexive Property of Congruence EXAMPLE 1 Use the SAS Congruence Postulate Write a proof. GIVEN PROVE

EXAMPLE 1 Use the SAS Congruence Postulate STATEMENTS REASONS ABCCDA SAS Congruence Postulate

Because they are vertical angles, PMQRMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MSare all equal. ANSWER MRSand MPQ are congruent by the SAS Congruence Postulate. EXAMPLE 2 Use SAS and properties of shapes In the diagram, QSand RPpass through the center Mof the circle. What can you conclude about MRSand MPQ? SOLUTION

Prove that SVRUVR STATEMENTS REASONS SV VU Given SVRRVU Definition of line Reflexive Property of Congruence RVVR SVRUVR SAS Congruence Postulate for Examples 1 and 2 GUIDED PRACTICE In the diagram, ABCDis a square with four congruent sides and four right angles. R, S, T, and Uare the midpoints of the sides of ABCD. Also, RT SUand . SU VU

STATEMENTS REASONS Given BS DU Definition of line RBSTDU Given RSUT SAS Congruence Postulate BSRDUT for Examples 1 and 2 GUIDED PRACTICE BSRDUT Prove that

Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.

WYZXZY PROVE 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 Write a proof. GIVEN WY XZ,WZ ZY, XY ZY SOLUTION

STATEMENTS REASONS WY XZ Given WZ ZY, XY ZY Given Definition of lines Z andY are right angles Definition of a right triangle WYZand XZY are right triangles. ZY YZ L Reflexive Property of Congruence WYZXZY HL Congruence Theorem EXAMPLE 3 Use the Hypotenuse-Leg 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 knowthatRP QS and PQ PS. What postulate or theorem can you use to conclude that PQRPSR?

You are given that PQ PS. By the Reflexive Property, RP RP. By the definition of perpendicular lines, both RPQ and RPSare 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 . PQRPSR EXAMPLE 4 Choose a postulate or theorem SOLUTION

Redraw ACBand DBCside by side with corresponding parts in the same position. for Examples 3 and 4 GUIDED PRACTICE Use the diagram at the right.

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

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

4.4 Prove triangles are congruent by SAS and HL