Chapter 5 Introduction to Trigonometry: 5.2 Congruent & Similar Triangles

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


Congruent, Similar or Neither?


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• 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)

XZ

AC

YZ

BC

XY

AB

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







• Congruent, similar or neither?


• 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 )≃





• 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 )≃


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

YZ

BC

XY

AB




• 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 ~)

YZ

BC

XY

AB





• 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 TrianglesEx. 1


• 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.)


• 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.)



FD

CA

EF

BC

DE

AB

6.25

2.19

16

12

20

15

75.075.075.0

So, yes, ∆ABC ~ ∆DEF because the ratio of the sides are the same so the sides are proportional

4

3

4

3

4

3



ZY

CA

XZ

BC

YX

AB

*10

15

8

12

6

9

5.15.15.1

So, yes, ∆ABC ~ ∆YXZ because the ratio of the sides are the same so the sides are proportional

2

3

2

3

2

3

*Found with pythagorean theorem



Homework

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

Documents

Chapter 5 Introduction to Trigonometry: 5.2 Congruent & Similar Triangles