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Proving Triangles Congruent

Proving Triangles Congruent. F. B. A. C. E. D. The Idea of a Congruence. Two geometric figures with exactly the same size and shape. How much do you need to know. . . . . . about two triangles to prove that they are congruent?.

joseph-rosario
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Proving Triangles Congruent

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  1. Proving Triangles Congruent

  2. F B A C E D The Idea of a Congruence Two geometric figures with exactly the same size and shape.

  3. How much do you need to know. . . . . . about two triangles to prove that they are congruent?

  4. Corresponding Parts • AB DE • BC EF • AC DF •  A  D •  B  E •  C  F B A C E F D In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. ABC DEF

  5. SSS SAS ASA AAS Do you need all six ? NO !

  6. Side-Side-Side (SSS) E B F A D C • AB DE • BC EF • AC DF ABC DEF Postulate 4-1: SSS Congruence Postulate:If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

  7. Side-Side-Side (SSS) Postulate 4-1 If all 3 sides of one triangle are congruent to all 3 sides of a second triangle, then the triangles are congruent.

  8. 2 – ( ) + 2 – ( ) x x y y 2 1 2 1 d = EXAMPLE 2 Standardized Test Practice SOLUTION By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR.

  9. 2 2 ( – ( ) ) – 1 ) (– 5 1 – 4 PR + = 2 2 = 5 ) (– 3 25 4 = + = By the SSS Congruence Postulate, any triangle with side lengths 3, 4, and 5 will be congruent to PQR. 2 2 2 2 = (–4) – (–1) ( ( 5 – 1) ) 5 ) (– 3 25 4 = + = + The correct answer is A. ANSWER EXAMPLE 2 Standardized Test Practice The distance from (–1, 1) to (–1, 5) is 4. The distance from (–1, 5) to (–4, 5) is 3. The distance from (– 1, 1) to (–4, 5) is

  10. Write a proof. GIVEN KL NL,KM NM PROVE KLMNLM Proof KL NL andKM NM It is given that LM LM. By the Reflexive Property, So, by the SSS Congruence Postulate, KLMNLM EXAMPLE 1 Use the SSS Congruence Postulate

  11. DFGHJK SideDG HK, SideDF JH,andSideFG JK. So by the SSS Congruence postulate, DFG HJK. for Example 1 GUIDED PRACTICE Decide whether the congruence statement is true. Explain your reasoning. SOLUTION Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. Yes. The statement is true.

  12. ACBCAD 2. GIVEN : BC AD ACBCAD PROVE : It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD. PROOF: for Example 1 GUIDED PRACTICE Decide whether the congruence statement is true. Explain your reasoning. SOLUTION

  13. Included Angle The angle between two sides H G I

  14. E Y S Included Angle Name the included angle: YE and ES ES and YS YS and YE E S Y

  15. Side-Angle-Side (SAS) Postulate 4-2 B E F A C D • AB DE • A D • AC DF ABC DEF included angle

  16. Side-Angle-Side (SAS) Postulate 4-2 If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

  17. C F, BC EF A D, GIVEN ABCDEF PROVE EXAMPLE 2 Prove the AAS Congruence Theorem Prove the Angle-Angle-Side Congruence Theorem. Write a proof.

  18. In the diagram at the right, what postulate or theorem can you use to prove that ? Explain. RSTVUT STATEMENTS REASONS Given S U Given RS UV The vertical angles are congruent RTSUTV for Examples 1 and 2 GUIDED PRACTICE SOLUTION

  19. Do NowEliminate the possibilities…. • Some of the measurements of ABC and DEF are given below. Can you determine if the two triangles are congruent from this information? 2.5 30 4cm 2.5 cm 30 4 cm

  20. Included Side The side between two angles GI GH HI

  21. E Y S Included Side Name the included angle: Y and E E and S S and Y YE ES SY

  22. Angle-Side-Angle (ASA) Postulate 21 B E F A C D • A D • AB  DE • B E ABC DEF included side

  23. Angle-Side-Angle (ASA) Postulate 4-3 If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

  24. Angle-Angle-Side (AAS) Theorem 4.6 B E F A C D • A D • B E • BC  EF ABC DEF Non-included side

  25. Angle-Angle-Side (AAS) • If two angles and a nonincluded side of a triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent. O A Y M N B

  26. Angle-Side-Side • THERE IS NO SUCH THING AS ANGLE SIDE SIDE BECAUSE YOU CAN’T USE THAT KIND OF LANGUAGE AT SCHOOL.

  27. Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT

  28. Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT

  29. Tell whether you can use the given information at determine whether • ABC   DEF • A  D, ABDE, ACDF • AB  EF, BC  FD, AC DE

  30. SSS correspondence • ASA correspondence • SAS correspondence • AAS correspondence • SSA correspondence • AAA correspondence The Congruence Postulates

  31. Name That Postulate (when possible) SAS ASA SSA SSS

  32. Name That Postulate (when possible) AAA ASA SSA SAS

  33. Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS

  34. HW: Name That Postulate (when possible)

  35. HW: Name That Postulate (when possible)

  36. Let’s Practice ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS:

  37. Closure Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:

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