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Geometry

Geometry. Triangle Congruence Theorems. Let’s Review. B. ~. =. A. C. Given : ABC DEF List the 6 congruency statements that can be concluded. E. D. F. Congruent Triangles. Congruent triangles have three congruent sides and three congruent angles.

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Geometry

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  1. Geometry Triangle Congruence Theorems

  2. Let’s Review B ~ = A C Given: ABC DEF List the 6 congruency statements that can be concluded. E D F

  3. Congruent Triangles • Congruent triangles have three congruent sides and three congruent angles. • However, we can prove triangles are congruent without showing that all 6 parts are congruent.

  4. Triangle Congruence Theorems ASA SAS AAS SSS All 3 sides congruent 2 Angles and a Non-included Side 2 Sides and the Included Angle 2 Angles and the Included Side

  5. 85° 30° 85° 30° • If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. DO YOU KNOW WHY? • Are 3 pairs of congruent angles enough to say for certain that these 2 triangles are congruent?

  6. 30° 30° Example These 2 triangles have congruent angles. Do you agree? Are the triangles congruent? What do we call figures that have the same shape but not the same size?

  7. So, how do we prove that two triangles really are congruent?

  8. A C B D F E ASA (Angle, Side, Angle) • If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . . then the 2 triangles are CONGRUENT!

  9. A C B D F E AAS (Angle, Angle, Side) • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . . then the 2 triangles are CONGRUENT!

  10. A C B D F E SAS (Side, Angle, Side) • If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . . then the 2 triangles are CONGRUENT!

  11. A C B D F E SSS (Side, Side, Side) • In two triangles, if 3 sides of one are congruent to three sides of the other, . . . then the 2 triangles are CONGRUENT!

  12. Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS – Pairs of congruent angles and the side not contained between them. SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides

  13. A C B D F E HL (Hypotenuse, Leg) • If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

  14. A C B Example 1 • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D D D D E E E E F F F F

  15. A C B D E F Example 2 • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

  16. A C B D Example 3 • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

  17. D F E A C B Example 4 • Why are the two triangles congruent? • What are the corresponding vertices? SAS A   D C   E B   F

  18. Example 5 A • Why are the two triangles congruent? • What are the corresponding vertices? D B SSS A   C ADB   CDB C ABD   CBD

  19. B C A D Example 6 • Given: Are the triangles congruent? S S S Why?

  20. Q P T R S Example 7 mQSR = mPRS = 90° • Given: • Are the Triangles Congruent? Why? R H S QSR  PRS = 90° This is the HL Postulate!

  21. THE END!!!

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