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Congruent Triangles

Congruent Triangles. We will… …name and label corresponding parts of congruent triangles. …use congruent triangle methods to show that two triangles are congruent. Corresponding parts of congruent triangles. Triangles that are the same size and shape are congruent triangles .

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Congruent Triangles

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  1. Congruent Triangles We will… …name and label corresponding parts of congruent triangles. …use congruent triangle methods to show that two triangles are congruent

  2. Corresponding parts of congruent triangles Triangles that are the same size and shape are congruent triangles. Each triangle has three angles and three sides. If all six corresponding parts are congruent, then the triangles are congruent.

  3. B Y Z C X A ~ ΔABC=ΔXYZ Corresponding parts of congruent triangles If ΔABC is congruent to ΔXYZ , then vertices of the two triangles correspond in the same order as the letter naming the triangles.

  4. B Y Z C X A ~ ΔABC=ΔXYZ Corresponding parts of congruent triangles This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles.

  5. Definition of Congruent Triangles (CPCTC) Two triangles are congruent if and only if their corresponding parts are congruent. CPCTC Corresponding Parts of Congruent Triangles are Congruent

  6. Answer:Since corresponding parts of congruent triangles are congruent, Example 3-1a ARCHITECTUREA tower roof is composed of congruent triangles all converging toward a point at the top. Name the corresponding congruent angles and sides of HIJ and LIK.

  7. The support beams on the fence form congruent triangles. a.Name the corresponding congruent angles and sides of ABC and DEF. b.Name the congruent triangles. Answer:ABC DEF Example 3-1c Answer:

  8. We don’t need to prove all 6 orresponding parts are congruent. • We have 5 short cuts or methods.

  9. Congruent Triangle Methods • SSS: Side-Side-Side • SAS: Side-Angle-Side • ASA: Angle- Side- Angle • AAS: Angle-Angle-Side • HL: Hypotenuse-Leg

  10. SSS • If we can show all 3 pairs of corresponding sides are congruent, the triangles have to be congruent

  11. Included angle SAS • Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent

  12. Which method can be used toprove the triangles are congruent

  13. Common sideSSS Vertical angles SAS Parallel lines alternate interior angles Common side SAS

  14. ASA – 2 angles and the included side • AAS – 2 angles and the non-included side S A A A S

  15. HL ( hypotenuse leg ) is usedonly with right triangles, BUT, not all right triangles. ASA HL

  16. Properties of Triangle Congruence Congruence of triangles is reflexive, symmetric, and transitive. REFLEXIVE ΔJKL = ΔJKL K K ~ L L J J

  17. Properties of Triangle Congruence Congruence of triangles is reflexive, symmetric, and transitive. SYMMETRIC ~ If ΔJKL = ΔPQR, then ΔPQR =ΔJKL. K Q L ~ R J P

  18. Properties of Triangle Congruence Congruence of triangles is reflexive, symmetric, and transitive. TRANSITIVE ~ If ΔJKL = ΔPQR, and ΔPQR = ΔXYZ, then ΔJKL =ΔXYZ. ~ K Q ~ L R J Y P Z X

  19. B E C F A D IDENTIFY CONGRUENCE TRANSFORMATIONS If you slide ΔABC down and to the right, it is still congruent to ΔDEF. B C A

  20. B E C F A D IDENTIFY CONGRUENCE TRANSFORMATIONS If you turn ΔABC, it is still congruent to ΔDEF. A B C

  21. B E C F A D IDENTIFY CONGRUENCE TRANSFORMATIONS If you flip ΔABC, it is still congruent to ΔDEF. A C B

  22. COORDINATE GEOMETRYThe vertices of RSTare R(─3, 0), S(0, 5), and T(1, 1). The vertices of RSTare R(3, 0), S(0, ─5), and T(─1, ─1). Verify that RST RST. Example 3-2a

  23. Example 3-2b Use the Distance Formula to find the length of each side of the triangles.

  24. Example 3-2b Use the Distance Formula to find the length of each side of the triangles.

  25. Example 3-2b Use the Distance Formula to find the length of each side of the triangles.

  26. TempCopy Answer: The lengths of the corresponding sides of two triangles are equal. Therefore, by the definition of congruence, In conclusion, because , Example 3-2c Use a protractor to measure the angles of the triangles. You will find that the measures are the same.

  27. Example 3-2d COORDINATE GEOMETRYThe vertices of RSTare R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Name the congruence transformation for RST and RST. Answer:RSTis a turn of RST.

  28. a. Verify that ABC ABC. Example 3-2f COORDINATE GEOMETRYThe vertices of ABC are A(–5, 5), B(0, 3), and C(–4, 1). The vertices of ABCare A(5, –5), B(0, –3), and C(4, –1). Answer: Use a protractor to verify that corresponding angles are congruent.

  29. Example 3-2g b.Name the congruence transformation for ABC and ABC. Answer:turn

  30. BOOKWORK: p. 195 #9 – 19, #22 – 25 (just name the congruence transformation) HOMEWORK: p.198 Practice Quiz

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