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

2. Humour Break

3. 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

4. 5.2 Congruent & Similar Triangles
Congruent, Similar or Neither?

5. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

6. 5.2 Congruent & Similar Triangles
If ΔABC is congruent (≃) with ΔXYZ
Corresponding sides must be equal, and
Corresponding angles must be equal

7. 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

8. 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

9. 5.2 Congruent & Similar Triangles
Therefore, ΔABC is ≃ with ΔXYZ

10. 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 ≃

11. 5.2 Congruent & Similar Triangles
Congruent, Similar or Neither?

12. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

13. 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)

14. 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)

15. 5.2 Congruent & Similar Triangles
Therefore, ΔABC is ~ (similar to) ΔXYZ because 3 pairs of corresponding angles are equal

16. 5.2 Congruent & Similar Triangles
Congruent, Similar or Neither?

17. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

18. 5.2 Congruent & Similar Triangles

19. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

20. 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 ≃)

21. 5.2 Congruent & Similar Triangles

22. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

23. 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 ≃)

24. 5.2 Congruent & Similar Triangles
If given that... AB:XY & BC:YZ are proportional, that is...

25. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

26. 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 ~)

27. 5.2 Congruent & Similar Triangles

28. 5.2 Congruent & Similar Triangles
Congruent, similar or neither?

29. 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 ~)

30. 5.2 Congruent & Similar Triangles
Ex. 1

31. 5.2 Congruent & Similar Triangles
In ∆ ABC, we can use the pythagorean theorem to find side AC
AC² = AB² + CB²
AC² = 15² + 12²
AC² =
AC² = 369
√AC² = √369
AC = 19.2 (approx.)

32. 5.2 Congruent & Similar Triangles
In ∆ DEF, we can use the pythagorean theorem to find side DF
DF² = DE² + EF²
AC² = 20² + 16²
AC² =
AC² = 656
√AC² = √656
AC = 25.6 (approx.)

33. 5.2 Congruent & Similar Triangles

34. 5.2 Congruent & Similar Triangles
So, yes, ∆ABC ~ ∆DEF because the ratio of the sides are the same so the sides are proportional

35. 5.2 Congruent & Similar Triangles
Ex. 2

36. 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

37. 5.2 Congruent & Similar Triangles
Ex. 3

38. 5.2 Congruent & Similar Triangles
Ex. 4

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