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Introduction To Congruent Triangles & Similar Triangles

Concept of congruence and similarity of triangles is explained.

Malakpreet
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Introduction To Congruent Triangles & Similar Triangles

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  1. Introduction To Congruent & Similar Triangles

  2. Goals for Today: • (1) Under stand the different between congruent figures and similar figures & in particular, congruent & similar triangles • (2) Understand how we can identify if two triangles are congruent or similar • (3) Understand how to find unknown measures or angles given two similar triangles

  3. Congruent, similar or neither?

  4. Congruent & Similar Triangles • If ΔABC is congruent (≃) with ΔXYZ •Corresponding sides must be equal, and •Corresponding angles must be equal

  5. • If ΔABC is congruent (≃) with ΔXYZ • Corresponding sides must be equal • Corresponding angles must be equal •AB = XY, BC = YZ and AC = XZ •So, the corresponding sides are equal

  6. • If ΔABC is congruent (≃) with ΔXYZ • Corresponding sides must be equal • Corresponding angles must be equal • A = X, B= Y and C= Z • So, the corresponding angles are equal

  7. Congruent & Similar Triangles • Therefore, ΔABC is ≃ with ΔXYZ

  8. • In fact, ΔABC is ≃ (congruent) with ΔXYZ if you can establish that corresponding sides are equal, that is: • AB = XY • BC = YZ • AC = XZ • You don’t have to measure the angles as well in this case, we have what is known as Side-side-side Congruence or SSS ≃

  9. Congruent, similar or neither?

  10. Congruent & Similar Triangles • If ΔABC is similar to ~ (similar) to ΔXYZ •Corresponding sides must be proportional (unlike congruent triangles where they must be equal), and •Corresponding angles must be equal (like congruent triangles)

  11. • If ΔABC is similar to (~) similar to ΔXYZ • Corresponding sides must be proportional AB BC AC   XY YZ XZ •Corresponding angles must be equal • A = X, B= Y and C= Z •If one or the other is established, the triangles are similar (you don’t have to prove both)

  12. • Therefore, ΔABC is ~ (similar to) ΔXYZ because 3 pairs of corresponding angles are equal

  13. • Congruent, similar or neither? • AB = XY and BC = YZ • B = Y • ∆ ABC ≃ (congruent) to ∆ XYZ • ∆ ABC ≃ (congruent) to ∆ XYZ if two pairs of corresponding sides and the contained angles are equal (SAS ≃)

  14. • Congruent, similar or neither? • BC = YZ • B = Y & C = Z • ∆ ABC ≃ (congruent) to ∆ XYZ • ∆ ABC ≃ (congruent) to ∆ XYZ if two pairs of corresponding angles and the contained side are equal (ASA ≃)

  15. If given that... AB:XY & BC:YZ are proportional, that is... AB BC XY YZ

  16. • Congruent, similar or neither? • B = Y & XY AB BC YZ • ∆ ABC ~ (similar) to ∆ XYZ • ∆ ABC ~ (similar) to ∆ XYZ if two pairs of corresponding sides are proportional and the contained angles are equal (SAS ~)

  17. • Congruent, similar or neither? • B = Y & C = Z • ∆ ABC ~ (similar) to ∆ XYZ • ∆ ABC ~ (similar) to ∆ XYZ if two pairs of corresponding sides are equal, then the third angles must also be angle and the triangles are similar (AA ~)

  18. Ex. 2

  19. *Found with pythagorean theorem 9 12 15   3 3 3 6 8 10 *   2 2 2 5 . 1  5 . 1  5 . 1 So, yes, ∆ABC ~ ∆YXZ because the ratio of the sides are the same so the sides are proportional

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