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You will use sides and angles to prove congruence.

4.5 Prove Triangles Congruent by SAS and HL. predict. 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 .

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You will use sides and angles to prove congruence.

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  1. 4.5 Prove Triangles Congruent by SAS and HL predict • 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.

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

  3. 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

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

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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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?

  11. 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

  12. 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.

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

  14. 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

  15. 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.

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

  17. State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate. 3. ST YZ, RS XY ANSWER SY. Daily Homework Quiz

  18. State a third congruence that would allow you to prove RSTXYZ by the SAS Congruence postulate. 4. T Z, RT XZ ANSWER STYZ . Daily Homework Quiz

  19. • Triangles are congruent by the SAS Congruence Postulate. • Right triangles are congruent by the HL Congruence Theorem. • 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 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.

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