Chapter 6
Proportions and
Similarity
SECTION 6.1:
Proportions
A RATIO is a comparison of two quantities.
Ratios can be expressed as:
𝒂
or a:b or “a to b”
𝒃
Extended Ratios can be applied to
3 or More Quantities (Like Triangles)
a:b:c or “a to b to c”
Sample Word Problems Involving Ratios and Triangles
1.If the Ratio of the
angles of a
triangle is 1:2:3,
then find the
measure of each
angle.
Sample Word Problems Involving Ratios and Triangles
1. If the Ratio of the angles of
a triangle is 1:2:3, then find
the measure of each angle.
• Solution:
• The angles of this triangle have to
follow this pattern – 1:2:3.
Meaning that the second angle is
double the first and the third is
triple the first.
• We also know that the angles
must add up to 180°.
• Therefore, represent the “multiplier”
with a variable, write an equation and
solve. Then substitute to find the
angles.
1x + 2x + 3x = 180
6x = 180
x = 30
Therefore the angles are:
1(30)= 30°, 2(30)= 60°, and 3(30)= 90°
Sample Word Problems Involving Ratios and Triangles
2. If the Ratio of the
sides of a triangle is
9:8:7, and the
perimeter is 144
inches. Find the
measure of each
side.
Sample Word Problems Involving Ratios and Triangles
2. If the Ratio of the
sides of a triangle is
9:8:7, and the
perimeter is 144
inches. Find the
measure of each
side.
• Since perimeter is the sum of the
sides, and we know the total we can
write an equation and solve as we
did before.
9x + 8x + 7x = 144
24x = 144
x=6
Therefore the sides are:
9(6) = 54 in, 8(6) = 48 in, 7(6) = 42 in.
Two equal Ratios form a PROPORTION
Solve Each of the Following:
3
5
1. =
2.
𝑦 −2
2
3.
𝑧 −1
𝑧+1
𝑥
75
=
4
5
=
5
6
Solve Each of the Following:
3
5
1. =
2.
𝑦 −2
2
3.
𝑧 −1
𝑧+1
𝑥
75
=
4
5
=
5
6
1. 5x = 3(75)
5x = 225
x = 45
2. 2(4) = 5(y – 2)
8 = 5y – 10
18 = 5y
3.6 = y
3. 6(z-1) = 5(z+1)
6z – 6 = 5z + 5
z = 11
SECTION 6.2:
Similar Polygons
When polygons are the same shape, but different
in size they are called SIMILAR, ~
Order MATTERS!!!!, We use Similarity Statements to
ID Corresponding parts
The 2 Polygons are SIMILAR. Write a Sim Statement,
find x, y and UT. Then ID the scale factor.
Sim Statement: RSTUV ~ ABCDE
UT = 20.5 + 2 = 22.5 m
SECTION 6.3:
Similar Triangles
There are 3 Rules for Proving Triangle Similarity
Determine whether the triangles are similar.
SECTION 6.4:
Parallel Lines and
Proportional Parts
𝑅𝐿
𝑅𝑊
• =
𝐿𝑇
𝑊𝑆
5
𝑥
• =
𝐿𝑇
?
6
5
𝑥
• =
4
6
• 4x = 30
• x = 7.5 u
= 𝑅𝑇 − 𝑅𝐿
𝑇𝑅
𝑆𝑅
• =
𝐿𝑅
𝑊𝑅
8
𝑥
• =
3
6
• 3x = 48
• x = 16
• WS = RS – RW
• WS = 10 u
Find x
Find x
• Due to the tick marks, we know
that the segment labelled “x” is a
midsegment.
• Therefore, it is half of the
segment it is parallel to.
• x = (1/2) 35
• x = 17.5 u
SECTION 6.5:
Parts of Similar
Triangles
If Two Triangles are Similar, then….
•Corresponding sides are proportional,
•Medians are proportional,
•Altitudes are proportional,
•Angle Bisectors are proportional, and
•The Perimeters of the Triangles are
Proportional to the sides.
So, the perimeter of ∆LMN = 35 units
Find x:
Find x:
• Solution:
20
•
7
=
24
𝑥
• 20x = 168
• x = 8.4 units