Chapter 5 Introduction to Trigonometry: 5 Chapter 5 Introduction to Trigonometry: 5.2 Congruent & Similar Triangles

Humour Break

5.2 Congruent & Similar Triangles 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

5.2 Congruent & Similar Triangles Congruent, Similar or Neither?

5.2 Congruent & Similar Triangles Congruent, similar or neither?

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

5.2 Congruent & Similar Triangles 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

5.2 Congruent & Similar Triangles 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

5.2 Congruent & Similar Triangles Therefore, ΔABC is ≃ with ΔXYZ

5.2 Congruent & Similar Triangles 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 ≃

5.2 Congruent & Similar Triangles Congruent, Similar or Neither?

5.2 Congruent & Similar Triangles Congruent, similar or neither?

5.2 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)

5.2 Congruent & Similar Triangles If ΔABC is similar to (~) similar to ΔXYZ Corresponding sides must be proportional 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)

5.2 Congruent & Similar Triangles Therefore, ΔABC is ~ (similar to) ΔXYZ because 3 pairs of corresponding angles are equal

5.2 Congruent & Similar Triangles Congruent, Similar or Neither?

5.2 Congruent & Similar Triangles Congruent, similar or neither?

5.2 Congruent & Similar Triangles

5.2 Congruent & Similar Triangles Congruent, similar or neither?

5.2 Congruent & Similar Triangles 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 ≃)

5.2 Congruent & Similar Triangles

5.2 Congruent & Similar Triangles Congruent, similar or neither?

5.2 Congruent & Similar Triangles 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 ≃)

5.2 Congruent & Similar Triangles If given that... AB:XY & BC:YZ are proportional, that is...

5.2 Congruent & Similar Triangles Congruent, similar or neither?

5.2 Congruent & Similar Triangles Congruent, similar or neither? B = Y & ∆ ABC ~ (similar) to ∆ XYZ ∆ ABC ~ (similar) to ∆ XYZ if two pairs of corresponding sides are proportional and the contained angles are equal (SAS ~)

5.2 Congruent & Similar Triangles

5.2 Congruent & Similar Triangles Congruent, similar or neither?

5.2 Congruent & Similar Triangles 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 ~)

5.2 Congruent & Similar Triangles Ex. 1

5.2 Congruent & Similar Triangles In ∆ ABC, we can use the pythagorean theorem to find side AC AC² = AB² + CB² AC² = 15² + 12² AC² = 225 + 144 AC² = 369 √AC² = √369 AC = 19.2 (approx.)

5.2 Congruent & Similar Triangles In ∆ DEF, we can use the pythagorean theorem to find side DF DF² = DE² + EF² AC² = 20² + 16² AC² = 400 + 256 AC² = 656 √AC² = √656 AC = 25.6 (approx.)

5.2 Congruent & Similar Triangles

5.2 Congruent & Similar Triangles So, yes, ∆ABC ~ ∆DEF because the ratio of the sides are the same so the sides are proportional

5.2 Congruent & Similar Triangles Ex. 2

5.2 Congruent & Similar Triangles *Found with pythagorean theorem So, yes, ∆ABC ~ ∆YXZ because the ratio of the sides are the same so the sides are proportional

5.2 Congruent & Similar Triangles Ex. 3

5.2 Congruent & Similar Triangles Ex. 4

Homework Wednesday, December 15th – page 460, #1-7