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Similarity of Triangles
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Definition: Triangle ABC is similar to triangle DEF if all corresponding angles are congruent, and all
pairs of corresponding sides are proportional. In this Example:
Angle A is congruent to Angle D
Angle B is congruent to Angle E
Angle C is congruent to Angle F
Segment AB is proportional to Segment DE
Segment AC is proportional to Segment DF
Segment BC is proportional to Segment EF
A
B
D
C
E
F
Proportional means the ratios of the three pairs of corresponding sides are all equal. For example,
AB = AC = BC
DE
DF
EF (or the reciprocals of each)
Note: If two triangles are congruent, then they are similar.
Similarity Properties of Triangles:
1) AA(A) Similarity: Two pairs of corresponding angles are congruent. (Note: if two pairs of
corresponding angles are congruent, then the third pair must be).
Example: At a certain time of day, a
flagpole casts a shadow of 40 feet.
At the same time, a student who is
5’6” tall, casts a shadow of eight feet.
How tall is the flagpole?
Solution: We know the triangles are
similar by AA because:
So, we can set up the following ratios
(after converting measurements to
common units):
a
5’6”
8 feet
40 feet
And solving for the unknown variable
(a) we get:
2) SSS Similarity: all three pairs of corresponding sides are proportional
Example: Devin wishes to build a scale
model of the Pyramids in Egypt. Each
side of the pyramid is an isosceles
triangle (basically) Suppose the three
sides of one of the triangles measure
1000 feet (base), 1300 feet and 1300
feet. Devin constructs his scale model
with a base of 8 inches. What are the
lengths of the other two sides?
Solution: Since we know the sides are
proportional, we can set up the
following ratios:
So, solving for the unknown variable,
we get:
© Dr Brian Beaudrie and Dr Barbara Boschmans
Similarity of Triangles
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3) SAS Similarity: two pairs of corresponding sides are proportional and their included angles are
congruent.
Example: If M is divides both segments
AC and BD such that AM is half of MC
Solution:
and BM is half of MD , with AM  8
inches BM  18 inches and AB  24
inches, find DC
B
A
M
C
D
Examples
1. Two hikers wanted to measure the distance across a canyon. They sighted two boulders
(called them A and B) on the opposite side of the canyon. From points directly across from
boulders A and B on their side of the canyon, they measured and found the distance to be 60
meters. Then one of the hikers stood at a point (C); from that point, the other hiker measured
the distances that form CDE drawn below. How far is it across the canyon?
A
60 meters
B
Canyon
D
E
30 meters
50 meters
40 meters
C
© Dr Brian Beaudrie and Dr Barbara Boschmans
Similarity of Triangles
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2. Answer Yes or No for each of the following questions.
a.
b.
c.
d.
e.
f.
g.
_____ All similar triangles are congruent triangles.
_____ All congruent triangles are similar triangles.
_____ All isosceles triangles are similar.
_____ All equilateral triangles are similar.
_____ All squares are similar.
_____ All rectangles are similar.
_____ All congruent polygons are similar.
3. Below is a map of Utah. If the
actual distance across the bottom
of the state is 275 miles, determine
the scale of the map:
4. For the lake below, explain
why AXB ~ DXC , if we
know that CD || AB .
1 inch = ___________ miles.
Then, determine the approximate lengths
of the other sides.
If AX = 180 feet, DX  90 feet, and
CD = 120 feet, what is the distance
across the lake?
5. Explain how you could use shadows and similarity to find the height of a building.
© Dr Brian Beaudrie and Dr Barbara Boschmans
Similarity of Triangles
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6. Which of the following pairs of triangles are similar? If they are similar, explain why, using one
of the three properties of similarity (AA, SSS, SAS). If they are not similar, explain why.
a.
35o
35o
70o
2
70o
3
b.
2
c.
8
6
2
6
2
7. Assume that the triangle is similar and find the measures of the unknown sides.
5
x
y
4
6
12
8. To find the height of a tree, a group of Girl Scouts devised the following method. A girl walks
towards the tree along its shadow until the shadow of the top of her head coincides with the
shadow of the top of the tree. If the girl is 150 cm tall, her distance to the base of the tree is 15
meters, and the length of her shadow is 3 meters, how tall is the tree?
© Dr Brian Beaudrie and Dr Barbara Boschmans