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Geometry

Geometry. Triangle Congruence Theorems. Congruent Triangles. Congruent triangles have three congruent sides and and three congruent angles. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles. FOR ALL TRIANGLES. SSS. ASA. AAS. SAS.

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Geometry

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  1. Geometry Triangle Congruence Theorems

  2. Congruent Triangles • Congruent triangles have three congruent sides and and three congruent angles. • However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles.

  3. FOR ALL TRIANGLES SSS ASA AAS SAS FOR RIGHT TRIANGLES ONLY HL The Triangle Congruence Postulates &Theorems

  4. Theorem • If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent. • Think about it… they have to add up to 180°.

  5. 85° 30° 85° 30° A closer look... • If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. • But do the two triangles have to be congruent?

  6. 30° 30° Example Why aren’t these triangles congruent? What do we call these triangles?

  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)Special case of ASA • 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. 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!

  13. 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 E F

  14. 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?

  15. 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?

  16. 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

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

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

  19. Q P T R S Example 7 mQSR = mPRS = 90° • Given: • Are the Triangles Congruent? Why? R H S QSR  PRS = 90°

  20. 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

  21. Summary ---for Right Triangles Only: HL – Pair of sides including the Hypotenuse and one Leg

  22. THE END!!!

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