Chapter 4 Congruent Triangles. 4.3 Proving Triangles Congruent: SSS and SAS.
Chapter 5 Introduction to Trigonometry: 5.2 Congruent & Similar Triangles.
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Transcript of 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
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)
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
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...
YZ
BC
XY
AB
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 ~)
YZ
BC
XY
AB
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 TrianglesEx. 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
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
5.2 Congruent & Similar TrianglesEx. 2
5.2 Congruent & Similar Triangles
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
5.2 Congruent & Similar TrianglesEx. 3
5.2 Congruent & Similar TrianglesEx. 4
Homework
• Wednesday, December 15th – page 460, #1-7