Download 11.1 Similar and Congruent Triangles

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11.1 Similar and Congruent Triangles
Introduction
Geometric shapes, also known as figures, are an important part of the study of geometry. Recognizing
and using congruent and similar shapes make calculations and design work easier. For example, in most
design work, rather than using different shapes, a few shapes are copied and used in different positions
and/or produced in different sizes to complete the design.
When a shape is obtained from another figure by means of enlargement or reduction, its size will be
different from the original one, but it remains the same shape as the original one.
Similar figures have the same shape and retain the same angle at corresponding vertices (congruent).
They may or may not have the same size, but the lengths of the corresponding sides will be in proportion
between the figures.
Congruent figures have sides with lengths that are in proportion and equal angle at corresponding vertices.
It is important to note that two figures can be similar, but not congruent; however, two figures cannot be
congruent and not similar.
Similar Triangles
Similar figures must have the same shape, but their sizes may be different.
Two equal-sided polygons are said to be similar if all the corresponding angles are equal in measure
and the corresponding sides are proportional in length.
The symbol for similar is “~”.
~
~
~
Each pair of polygons shown are similar.
When writing the similarity relationship, the order in which the letters are written to represent the
similar figures is very important.
In similar figures, the measures of corresponding angles are equal and the ratio of the lengths of
corresponding sides are equal. The length of ratio between the corresponding sides of similar figures
are expressed as a fraction and is called the “scale” or scale factor.
For example, ∆ABC is similar to ∆DEF,
D
A
2
B
100°
50°
3
30°
4
(i) ∆ABC ~ ∆DEF
~
C
100°
4
×2
E
6
50°
30°
8
F
∠A = ∠ D, ∠ B =∠ E, and ∠C = ∠ F
(Corresponding angle measures of two triangles are equal.)
(ii)
∆ABC ~ ∆DEF
AB BC AC 2
=
=
=
DE EF DF 1
(Corresponding side lengths of two triangles are proportional.)
11.1 Similar and Congruent Triangles
380
Any triangle is defined by six measures: three sides and three angles. However, it is not necessary
to know all of the six measures to demonstrate that the two triangles are similar. If any one of the
following four conditions are met, then the two triangles are similar:
1. AAA (angle, angle, angle)
If all three pairs of corresponding angle measures of two triangles are equal, then the triangles are similar.
D
.
.
A
c
f
b
B
If ∠ A = ∠ D,
E
C
a
∠ B = ∠ E , and
∠ C = ∠ F,
e
then ∆ABC + ∆DEF
F
d
This is the same as AA (angle, angle) because if any two angles of the two triangles are equal, then
the third angle must be equal.
2. SSS (side, side, side)
If all three pairs of corresponding side lengths of two triangles are in the same proportion, then the triangles are similar.
D
A
f
b
c
F
C
a
B
d
E
BC AC
=
=
EF
DF
a
b
=
=
ci.e.,
d
e
If
e
AB
,
DE
c
m
f
then ∆ABC ~ ∆DEF
3. SAS (side, angle, side)
If two triangles have a pair of equal angle measures and the corresponding sides containing the
equal angles are in the same proportion, then the triangles are similar.
If ∠ B = ∠ E ,
D
A
B
e
f
b
c
C
a
E
AB BC
=
DE EF
c
a
i.e.,
=
f
d
then ∆ABC ~ ∆DEF
and
F
d
4. RHS (Right angle, hypotenuse,
side)
B
In right triangles, if the hypotenuses and lengths of one pair of corresponding sides are
B
proportional, then the triangles are similar.
D
If ∠B = ∠E = 90°, and
A
c
b
a
B
AC BC i.e., b = a ,
,
=
DF EF
e
d
then ∆ABC ~ ∆DEF
C
E
Example 11.1-a
and
e
f
F
d
Rules for Similar Triangles
Based on the information given in the following pairs of triangles, state the property that will prove
that they are similar.
(i)
(ii)
8
4
3
6
Solution
Chapter 11 | Basic Trigonometry
(i) SSS Property
4
2
(iii)
45°
30°
3
2
6
4
(ii) SAS Property
45°
30°
(iii)AAA Property
381
Example 11.1-b
Using Similar Triangles to Find the Unknown Length
(i) If ∆PQR ~ ∆XYZ, find XY and XZ
(ii) If ∆ABC ~ ∆DEF, find AB and DF
D
X
10
A
C
P
5
3
Q
B
Y
R
4
Therefore,
PQ QR PR
=
=
XY YZ XZ
333 444 555
==== ===
XY
XY
XY 888 XZ
XZ
XZ
4(XY) = 3(8)
Example 11.1-c
AB
AB
AB 666 1010
10
=== ====
1515
15=1212
12 DF
DF
DF
AB
AB 66 10
10
AB
AB 66 1010
== == == ===
15
15 12
12 DF
DF
1515 =1212 DF
DF
4(XZ) = 5(8)
3(8)
5(8)
= 6 XZ =
= 10
4
4
=6
= 10
Therefore, XY = 6 and XZ = 10.
XY =
F
12
(iii)∆ABC ~ ∆DEF
AB BC AC
Therefore,
=
=
DE EF DF
33 44 55 33 44 55
=== == == ==
XY
XY 88 XZ
XZXY
XY 88 XZ
XZ
E
Z
8
∆PQR ~ ∆XYZ
Solution
15
6
6(DF) = 10(12)
12(AB) = 6(15)
6(15)
= 7.5
12
= 7.5
AB =
10(12)
= 20
6
= 20
DF =
Therefore, AB = 7.5 and DF = 20.
Identifying Similar Triangles
Determine whether the following pairs of triangles are similar:
(i)
P
Y
6
20°
120°
Q
(ii)
9
40°
X
12
Z
R
18
A
D
9
B
Solution
(i)
3
50°
12
C
E
50°
F
4
∠ Q = 120°, ∠ Y = 180 – (40 + 20) = 120°
P
∠ Q = ∠ Y (Equal angles)
PQ 12
= = 2,
XY 6
6
X
12
Y
120°
9
120°
Q
18
R
Z
QR 18
= =2
YZ 9
PQ QR
(Corresponding sides containing the equal angles are in the same proportion)
=
YZ XY
∆PQR ~ ∆XYZ (SAS Property)
i.e.
11.1 Similar and Congruent Triangles
382
Solution
(ii)
continued
∠ B = ∠ E (Equal angle measures)
A
D
9
BC 12
= =3
EF 4
3
50°
B
C
12
E
50°
F
4
AC 9
= =3
DF 3
BC AC
i.e.,
(Corresponding
two side lengths are in the same proportion)
=2 =
2
EF
DF
However, the sides that are proportional do not contain the equal angles. Therefore, we cannot
conclude that ∆ABC is similar to ∆DEF.
Note: SSA (side, side, angle) is not sufficient to conclude that two triangles are similar.
Congruent Triangles
Congruent figures must have the same shape and size.
Two equal-sided polygons are said to be congruent if all the corresponding angles are equal in measure and
the corresponding sides are equal in length (i.e., the polygons are similar and they have equal side lengths).
The symbol for ‘congruent’ is ‘ ≅ ’.
≅
≅
≅
Each of the above pairs of polygons is congruent.
A
D
∆ABC ≅ ∆DEF
B
C
E
F
Once again, it is important to note that when the congruent relationship is written, the order of letters
representing the figures must be consistent to illustrate the equal corresponding angle measures and side
lengths. For example:
∆ABC ≅ ∆DEF
∠A = ∠D
(ii) ∠B = ∠E
(iii) ∠C = ∠F
(i)
ABC ≅ DEF
(i) AB = DE
(ii) BC = EF
(iii)AC = DF
Corresponding angle measures and side lengths are equal (congruent).
If any of the following four conditions are met, then the two triangles are congruent.
The condition of congruence for triangles requires all three sides and all three angle measures to be congruent.
However, it is not necessary to know all of the six measurements to demonstrate that the two triangles are
congruent.
1. SSS (side, side, side)
If all three pairs of corresponding side lengths of two triangles are equal, then the triangles are congruent.
A
B
Chapter 11 | Basic Trigonometry
D
C
E
If AB = DE,
BC = EF, and
AC = DF,
F
then ∆ABC ≅ ∆DEF
383
2. SAS (side, angle, side)
If two side lengths and the contained angle measure of one triangle is correspondingly equal to two
side lenghts and the contained angle measure of another triangle, then the triangles are congruent.
B
E
C
BC = EF, and
∠B = ∠E,
then ∆ABC ≅ ∆DEF
F
A
If AB = DE,
D
3. ASA (angle, side, angle)
If two angle measures and the contained side length of one triangle is correspondingly equal to two
angle measures and the contained side length of another triangle, then the triangles are congruent.
C
F
A
B
∠B = ∠E,
∠C = ∠F, and
BC = EF,
If
D
E
then ∆ABC ≅ ∆DEF
4. RHS (Right angle, hypotenuse, side)
In a right triangle, if the hypotenuse and side length of one triangle is correspondingly equal to the
B the triangles are congruent.
B
hypotenuse and side length
of another right triangle, then
D
A
∆ABC and DEF are right
triangle with ∠B = ∠E = 90°,
Therefore, if AC = DF, and
BC = EF,
then ∆ABC ≅ ∆DEF
B
Example 11.1-d
E
C
F
Determining Similarity/Congruency of Triangles
In the following examples, determine whether the pairs of triangles are similar, congruent, or neither:
(i)
(ii)
A
12 cm
N
7.5 cm
18 cm
5 cm
B
M
D
O
E
C
P
F
Q
(iii)
(iv)
J
G
L
T
4 in
H
Solution
U
I
AB
W
8 in
35°
8 in
4 in
V
X
35°
Y
K
BC
and the angle measures between those two sides ( ∠B and ∠D) are equal,
ED DF
both right triangles, ∆ABC and ∆EDF, are similar by the SAS Property (i.e., ∆ABC ~ ∆EDF).
(i) Since
=
Since the corresponding lengths of the sides are different, the triangles are not congruent.
11.1 Similar and Congruent Triangles
384
Solution
continued
(ii) Since MN and PQ are parallel, we know that alternate angles are equal (i.e., ∠N = ∠P and
∠M = ∠Q), and therefore, ∆MNO and ∆QPO are similar by the AA Property (i.e., ∆MNO ~
∆QPO).
Since the lengths of the sides are not known, we cannot determine congruency. However,
judging by the scale, it appears that they are not congruent.
(iii)Since GH = JK, GI = JL, and HI = KL, the triangles ∆GHI and ∆JKL are congruent by the SSS
Property. Since the triangles are congruent, they are similar.
(iv) Since UT = XW, TV = WY, and ∠V = ∠Y, it may be tempting to label ∆TVU and ∆WYX as
similar triangles. However, it is obvious that UV ≠ XY, and therefore, they are not in proportion
with the lengths of the other two sides; thus, the triangles are not similar (this is an example to
demonstrate that the SSA criterion is not sufficient to demonstrate similarity).
Since the corresponding pair of equal angles ( ∠V = ∠Y = 35°) is not contained by the
corresponding pair of equal sides of the triangles, the tirangles are not congruent.
Therefore, we conclude that ∆TUV and ∆WXY are not similar triangles. They are neither
similar nor congruent.
Example 11.1-e
Calculating Angle and Side Measures of Similar Triangles
Calculate the lengths of the unknown sides and unknown angle measures of the following pairs of
similar triangles:
(i)
D
A
7 cm
B
28°
45°
16 cm
18 cm
C
28°
E
(ii)
M
3.5
57°
45°
24 cm
F
2.5
78°
N
3.0
O
P
4.0
Q
Solution
(i) Since ∆ABC ~ ∆DEF, the corresponding side lengths of the two triangles are proportional.
AB
AC
BC
=
=
DF
DE EF
AB 16
7
i.e.,
=
=
18
24 DF
AB
16
=
18
24
16(18)
24
= 12.0 cm
AB =
Chapter 11 | Basic Trigonometry
16
7
=
24
DF
7(24)
DF =
16
= 10.5 cm
385
Solution
Since ∆ABC ~ ∆DEF, the corresponding angles of the two triangles are equal.
continued
∠C = ∠F = 45°, ∠E = ∠B = 28°
Therefore, ∠A = ∠D = 180° – 28° – 45° = 107°.
(ii) Since MN || PQ, ∠N = ∠P and ∠M = ∠Q, because they are alternate angles.
∠P = 78° and ∠Q = 57°. Therefore, ∆MNO ~ ∆QPO by the AA Property.
Hence, the corresponding side lengths of the triangles are proportional:
NO
MO
MN
=
=
.
QP
PO
QO
3.0
3.5
2.5
i.e.,
=
=
4.0
PO
QO
3.0
3.5
2.5
=
=
4.0
PO
QO
3.5
2.5
=
4.0
QO
4.0(3.5)
4.0(3.0)
QO =
2.5
2.5
= 5.6 in
= 4.8 in
Since the corresponding two angles of the triangles are known, the third angle can be computed as
follows: ∠MON = ∠QOP = 180° – (78° + 57°) = 45°. They are opposite angles as well (the angles
cannot simply be referred as ∠O, as that creates ambiguity).
PO =
Solving Application Problems using Similar and Congruent
Triangles
Similar and/or congruent triangles can be used to solve a variety of real-life application problems
when it is difficult or impossible to calculate certain angles or lengths.
Example 11.1-f
Determining the Height of a Building using Similar Triangles
The new science building at a college is 6 stories tall. Arianna wishes to know the height of the building.
She devises a method whereby, she and a friend measure the length of the shadow that the building casts
at 3:00 in the afternoon. The length of this shadow is 6.24 m. She then has her friend measure her height 165 cm - and the length of her shadow - 44 cm. Using this information, how can Arianna determine
the height of the science building?
Since the sun meets all points on the ground in a close vicinity at the same angle at any given time, and
it hits both the building and Arianna (standing vertically) at the same angle, the shadows created by
the building and Arianna form similar triangles, as shown in the diagram.
Solution
A
h
D
165 cm
B
6.24 m
C
E 44 cm F
Therefore, the measurements of the building’s height and length of its shadow are proportional to
Arianna’s respective measurements:
=
h
165 cm
=
6.24
m
44 cm
=2
165 cm
Solving this ratio for h, h = (6.24 m) S 44 cm X = 23.4 m.
Therefore, the science building has a height of approximately 23.4 m.
11.1 Similar and Congruent Triangles
386
Calculating the Distance across a Lake using Congruent Triangles
Example 11.1-g
A lake is situated on a property in the country. A couple looking to purchase the property wishes to
know how long the lake is. How can they determine this (without getting wet)?
They can each mark a point on either end of the lake (denoted A and B on the diagram below), each
a fixed distance away from the edge of the lake, and measure the distance to a common point on one
of the adjacent sides of the lake (denoted C). They can then each continue to walk the same distance
again in the same direction to another set of points on their property (denoted D and E respectively),
creating congruent triangles (by SAS). Then, they can measure the distance from D to E, which will
be the same distance as A to B, since the triangles are congruent. Finally, by subtracting the distance
from each of A and B to the edge of the lake, they will have determined the length of the lake.
Solution
A
B
C
D
E
11.1 Exercises
Answers to odd numbered problems are available at the end of the textbook.
In Problems 1 and 2, for each pair of similar triangles, name the proportional sides, and congruent angles.
1.
a. ∆ABC~ ∆DEF
b. ∆PQR ~ ∆RST
2.
a. ∆UVW ~ ∆XYZ
b. ∆GHI ~ ∆JKL
In Problems 3 and 4, for each pair of congruent triangles, name the equal side, and equal angles.
3.
a. ∆ABC , ∆XYZ
b. ∆DEF , ∆RST
4.
a. ∆UVW , JKL
b. ∆XYZ , ∆JKL
For Problems 5 to 10, identify the pairs of similar triangles and state the rule used to determine similarity.
5.
a.
4
6
Grb.
ad
5
7
8
6.
a.
rc.
12
9
9
Grb.
ad
70°
rc.
60 °
70 °
45°
70°
7.
a.
6
10
Grb.
ad
rc.
3
8
5
8
Chapter 11 | Basic Trigonometry
65°
6
387
8.
a.
4
7
Grb.
ad
rc.
7
9.
a.
60°
Grb.
7.5
ad
9 18°
3
10.
rc.
10
2 92°
60°
6
15
70°
a.
12
Grb.
ad
rc.
15
12
10
8
10
For Problems 11 to 16, identify the pair of congruent triangles and state the rule used to determine congruency.
11.
a.
Grb.
ad
rc.
12.
a.
Grb.
ad
rc.
13.
a.
Grb.
ad
rc.
14.
a.
Grb.
ad
rc.
15.
a.
Grb.
ad
rc.
16.
a.
Grb.
ad
rc.
11.1 Similar and Congruent Triangles
388
For Problems 17 to 22, determine whether each pair of triangles is congruent, similar, or neither:
B
17.
18.
10
8
7
5
14
16
15
7
11
12
9
5
B
19.
13.5
6
18
8
B
20.
5
21.
7
45°
B
5
45°
B
7
22.
13
In Problems 23 to 26, triangles and quadrilaterals are divided into two triangular pieces. Determine if the resulting pieces are
B
congruent, similar, or neither.
B
24.
23.
30°
100°
30°
100°
B
25.
26.
7.5 cm
In Problems 27 to 30, solve each pair of similar triangles completely, calculating
the unknown side lengths (rounded to the
B
nearest tenth as necessary) and angle measures.
27.
B
B
83°
4
A
28.
G
5
32°
5
A
41°
x
9
39°
C
D
I
J
9
41°
y
E
83°
K
6
H
15
6
x
39°
32°
L
y
F
B
29.
5
x
O
4
P
60°
B
30.
M
y
w
U
45°
45°
Q
V
x
z
y
8
Chapter 11 | Basic Trigonometry
T 45° 20
N
7
S
9
15
25°
25°
25°
R
389
31. A flagpole casts a 3.4 m shadow. Melanie, who is 160 cm tall, stands beside the flagpole, and her shadow is 64 cm long.
Draw a diagram and calculate the height of the flagpole.
32. A tree casts a 4.5m shadow. At the same time, a stick 55 cm long casts a shadow 90 cm long. Calculate the height of
B
the tree in metres and centimeters.
33. In an outdoor theatre at night, a spotlight is placed 8 m behind a
sheet that is 5.5 m high. As an actress, who is 1.65 m tall, walks
from the sheet towards the spotlight, it casts her shadow onto the
wall. How far away from the spotlight is the actress when her
shadow is the entire height of the sheet?
5.5 m
1.65 m
x
B
8m
34. A streetlight situated 7.4 m above the street casts a shadow on a
pedestrian, taking a late-night walk. If the pedestrian is 1.8 m tall,
how long is his shadow when he is 7 m away from the streetlight?
7.4 m
1.8 m
B
7m
x
35. A man is standing 12 m away from a lamp post which is 10 m high.
If his shadow is 2.2 m long, how tall is he?
10 m
h
12 m
36.
Two ladders of lengths 4 m and 9 m are leaning at the same angle
against a wall. If the 4 m ladder reaches 3.2 m up the wall, how
much further up the wall does the 9 m ladder reach?
B
2.2 m
y
9m
3.2 m
4m
B
37. Calculate the height of “y” in the diagram below.
38. Calculate the length of “x” in the diagram
below.
B
9m
15 m
12 m
6m
x
y
21 m
3m
11.1 Similar and Congruent Triangles