Download Congruency and Similarity

Unit 5 ‐ Trigonometry will include:
Similar Triangles vs. Congruent Triangles
‐scale factor
Trigonometric Ratios
‐Sine, Cosine, Tangent
‐relationship to slope
Sine Law
Cosine Law
Solving Triangles
Applications
Congruency and Similarity
Which of the following triangles are congruent?
Which are similar?
What are "corresponding sides"
and
"corresponding angles"?
1
Congruent Triangles
Triangles are CONGRUENT if they are identical in every way.
ΔABC ≅ ΔXYZ
"ΔABC is congruent to ΔXYZ"
Note: The order in which we write the vertices is important.
To be considered congruent, two triangles must satisfy
one of the following requirements:
(SSS≅)
Three pairs of corresponding sides are
equal.
(SAS≅)
Two pairs of corresponding sides and
the contained angle are equal.
(ASA≅)
Two pairs of angles and the contained
side are equal.
Similar Triangles
Triangles are SIMILAR if they have the same shape but are different sizes.
X
A
6
5
3
B
4
10
Y
C
8
Z
ΔABC ~ ΔXYZ "ΔABC is similar to ΔXYZ"
Note: The order in which we write the vertices is important.
In similar triangles,
(1) corresponding angles are equal and (2) corresponding (matching) sides are in the same ratio.
A
B
C
X
Y
Z
AB
XY
BC
YZ
AC
XZ
This factor is related
to the magnification!
Note that if one these conditions is true, then the other is also true.
2
Sufficient conditions for similarity
To show two triangles are similar, it is sufficient to show that one of the following is true:
i) the 3 pairs of sides are in the same ratio (SSS~).
ex.
P
6
5
3
R
10
Q T
4
8
∴ΔPRQ ~ ΔSTU (SSS~)
U
• Identify corresponding sides.
• Check the ratios between corresponding sides.
...Sufficient conditions for similarity
ii) two pairs of sides are in the same ratio and the L
contained angles are equal (SAS~).
L
10
ex.
10
5
M
8
P
M
5
N
N
4
Q
∴ΔPLQ ~ ΔMLN (SAS~)
• Identify corresponding sides.
• Check the ratios between corresponding sides.
• Check that the contained angles are equal.
3
...Sufficient conditions for similarity
iii) two pairs of angles are equal (AA~).
ex.
F
A
B
E
D
C
• Identify corresponding angles.
Similar Triangles Summary
If ΔABC ~ ΔXYZ, then
(SSS~)
all corresponding sides are proportional
(SAS~)
two pairs of corresponding sides are
proportional, and the contained angles are
equal.
(AA~)
two pairs of corresponding angles are
equal.
4
ex. Determine the value of m, n.
What triangles are similar?
B
m
A
3
C
E
6
4
n
5
D
Therefore, the values of m and n are 10/3 and 9/2 respectively.
Congruent triangles are identical in every way. They have
exact the same size and shape. They may be rotated.
Similar triangles have the same shape, but are different
sizes. One triangle is an enlargement or a reduction of
the other.
See p. 457 for a summary
Triangles
Similar
Congruent
5
HW: p. 460 # 1, 2, 3, 4iii
p. 539 # 3
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