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UNIT 4B: SIMILARITY
Common Core Standards
SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are
similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
SRT.4: Prove theorems about triangles: Side-Splitter Theorem, Pythagorean Theorem.
SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Vocabulary
Pythagorean Theorem
similarity ratio
ratio
indirect measurement
proportion
scale drawing
Cross-Products Property
dilation
similar polygons
UNIT 4B: SIMILARITY
Lesson 1 – Triangle Relationships

I can use the triangle inequality theorem

I can understand angle-side relationships in triangles.
Can any three sides form a triangle?
What is the relationship between the angles and sides of a triangle?
Examples:
1. Write the sides in order from shortest to
longest.
2. Write the angles in order from smallest to
largest. Is this is an acute, obtuse, or right
triangle? Verify your response.
3. Determine whether a triangle can have the given side lengths. Show work or explain your
reasoning. If it does form a triangle, determine if the triangle is right, obtuse, or acute.
a. 7, 10, 19
b. 2.3, 3.1, 4.6
c. 8, 13, 21
d. 7, 24, 25
4. A triangle has side lengths 7 and 12. What is the range of possible side lengths for the other side?
UNIT 4B: SIMILARITY
Lesson 2 – Ratios and Proportions

I can express a ratio in multiple formats.

I can solve a proportion.
A ratio compares two numbers by division. It can be written as:
A proportion is an equation stating two ratios are equal.
Properties of Ratios
The proportion
a c
 is equivalent to:
b d
Cross Products Property:
Examples:
Solving a proportion
Ex.
7 56

x 72
b.
3 5

x 14
c.
2x  3 x

5
4
d. BONUS:
 x  3  x
x
4
UNIT 4B: SIMILARITY
1. Given that 6 x  5 y after cross-multiplication, then
2. Given that
a.
g

f
x
=?
y
f
3
, complete each equation.

g 10
b.
3

f
c. 3g 
Proportional Relationships
3. According to a recent study, 7 out of 8 high school students have a cell phone. If there are 1273
students at Gull Lake High School, approximately how many of them have a cell phone?
4. The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of
each side?
5. Let x be the length of the model in centimeters. The rectangular model of the racing car is similar to
the rectangular racing car, so the corresponding lengths are proportional.
Find the length of the model to the nearest tenth of a centimeter.
Finding the height of an object using its shadow.
Tyler wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow
and then made a diagram. What is the height h of the pole?
UNIT 4B: SIMILARITY
Lesson 3 – Similar Figures

I can explain what it means for figures to be similar.

I can write a similarity statement.

I can find the similarity ratio (scale factor) for similar

I can verify that figures are similar.

I can find missing parts of similar figures.
figures.
We have already learned what it means for two figures to be congruent.
We are going to see what it means for two figures to be similar.
I2
B
C
X
B
C
Y
Y
X
D
A
N
A
D Z
W
Z
W
Two figures are congruent if and only if they
have the same
and
.
In other words,
Two figures are congruent if and only if
Two figures are similar if and only if they have
the same
, but not necessarily the
same
.
In other words,
Two figures are similar if and only if
.
.
Congruence Transformations:
Similarity Transformations:
Also, if two figures are congruent, then their
corresponding angles are
and
their corresponding side lengths are
.
Also, if two figures are similar then their
corresponding angles are
their corresponding side lengths are
.
The similarity ratio is
and
.
The similarity ratio of ABC to DEF is
. The scale factor is
.
The similarity ratio of DEF to ABC is
. The scale factor is
.
When figures are congruent to each other their similarity ratio is
.
UNIT 4B: SIMILARITY
Examples:
Determine if each pair of figures is similar to each other.
If so, identify the corresponding sides and angles and determine the similarity ratio and similarity
statement. If not, explain why they are not similar.
C
14
B
20
11
W
A
87°
14
V
10.5
D
15
8.25
20
U
87°
E
X
10.5
15
Y
0.75
B
120°
24
C
80°
20
11.9
A
B
110°
50°
32.4
10
D
A
120°
15
C
80°
110°
50°
19.4
D
5.4
UNIT 4B: SIMILARITY
Lesson 4 – Similar Triangles

I can verify that triangles are similar using AA~, SAS~, and SSS~, HL~

I can write a similarity statement.
Examples:
Prove or explain why the triangles are similar and write a similarity statement.
D
A
7.8
B
10
19.5
25
C
E
F
UNIT 4B: SIMILARITY
Refer to the diagram to the right.
Explain why ABE ~ ACD .
Now that you know the triangles are similar, determine CD.
Refer to the diagram at the right.
Explain why ABE ~ ACD .
Now that you know the triangles are similar, determine BE and
CD.
UNIT 4B: SIMILARITY
Lesson 5 – Other Proportional Relationships

I can apply the Side-Splitter Theorem and Converse of Side-Splitter Theorem

I can apply the Two-Transversal Proportionality Theorem.

I can apply the Triangle Angle Bisector Theorem.
Using the theorems involving proportional relationships, you can complete the following examples.
Examples:
In each example, state the theorem or reasoning you used.
Find SU.
Find PN.
If UV = 16, then find ST.
If PQ = 9, then find LM.
Is DE BC ? Verify with work and theorem.
Given that AC  36 , what value of BC would make
DE AB ? Verify with work and theorem.
UNIT 4B: SIMILARITY
Find LM and MN.
Find PS and SR.
Find the value of x.
UNIT 4B: SIMILARITY
Lesson 6 – Area and Perimeter of Similar Figures

I can solve real-world problems using similar figures
Area and Perimeters
When an object’s dimensions change proportionally, what happens to its perimeter? What
happens to its area?
Consider the following rectangles…
∆DEF is a dilation of ∆ABC, and BC = 4 mm, and EF = 12 mm. If the perimeter is 42 mm and
the area is 99 mm2 for ∆DEF, find the perimeter and area of ∆ABC.
E
B
3
D
4 mm
12 mm
A
C
D
F