Geometry – Chapter 7 – Notes and Examples
Sections 1 & 2 Ratios and Proportions and Ratios in Similar Polygons
A _________________ compares two numbers by ___________________. A ratio can be written as
_____________, ___________ or ____________ where b ≠ 0.
The slope of a line can be expressed as a ratio.
A ratio can involve more than two numbers such as the side lengths of a
rectangle can be shown as ____________________.
Problem 1
Given that two points on line l are A(–1, 3) and
B(2, –2), write a ratio expressing the slope of line l.
Problem 2
The ratio of the side lengths of a triangle is 4:7:5,
and its perimeter is 96 cm. What is the length of the
shortest side?
Problem 3
The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle?
A ______________________is an equation stating that two ratios are equal. In the proportion
𝑎
𝑏
=
𝑐
𝑑
, the
values _____ and _____ are the extremes. The values _____ and _____ are the means. When the proportion
is written as ________ = ________, the extremes are in the first and last positions. The means are in the
two middle positions.
In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of
the means bc are called the cross products.
Problem 4
Solve the proportion.
7
𝑥
=
56
3
72
8
Problem 6
Solve the proportion.
2𝑦
9
Problem 5
Solve the proportion.
=
=
𝑥
64
Problem 7
Solve the proportion.
8
𝑧−4
4𝑦
5
=
20
𝑧−4
Problem 8
Given that 18c = 24d, find the ratio of d to c in
simplest form. Hint: use Properties of Proportions.
Problem 9
Given that 16s = 20t, find the ratio t:s in simplest
form. Hint: use Properties of Proportions.
Figures that are ____________________ have the same shape but not necessarily the same size. The
symbol for similar is ________.
Problem 10
Identify the pairs of congruent angles and
corresponding sides.
Problem 11
Identify the pairs of congruent angles and
corresponding sides.
A __________________ ____________is the ratio of the lengths of the corresponding sides of two similar
polygons.
3
1
6
2
The similarity ratio of ∆ABC to ∆DEF is , or
6
. The similarity ratio of ∆DEF to ∆ABC is , or 2.
3
Writing a similarity statement is like writing a congruence statement — be sure to list corresponding
vertices in the same order.
Problem 12
Determine whether the polygons are similar. If
so, write the similarity ratio and a similarity
statement.
Step 1: Identify pairs of
congruent angles.
Step 2: Compare corresponding angles.
Problem 13
Determine if ∆JLM ~ ∆NPS. If so, write the
similarity ratio and a similarity statement.
Step 1: Identify
pairs of congruent
angles.
Step 2 Compare corresponding sides
Geometry – Chapter 7 – Notes and Examples
Section 3 – Triangle Similarity: AA, SSS, and SAS
There are several ways to prove certain triangles are similar. These ways are:
Hypothesis
Angle-Angle Similarity AA ~
If two angles of one triangle are
congruent o two angles of
another triangle, then the
triangles are similar.
Side-Side-Side Similarity SSS~
If the three sides of one triangle
are proportional to the three
corresponding sides of another
triangle, then the triangles are
similar.
Side-Angle-Side Similarity
SAS ~
If two sides of one triangle are
proportional to two sides of
another triangle and their
included angles are congruent,
then the triangles are similar
Problem 1
Problem 2
Explain why the triangles are similar and write a Verify that the
similarity statement.
triangles are
similar.
Conclusion
Problem 3
Verify that the triangles are similar.
Problem 4
Explain why ∆RSV ~ ∆RTU and then find RT.
Geometry – Chapter 7 – Notes and Examples
Section 4 Applying Properties of Similar Triangles
Problem 1
Find US.
Problem 2
Find PN.
Problem 3
Problem 4
Verify that DE ∥BC.
Verify that DE ∥AB.
Problem 5
Use the diagram to find LM and MN to the nearest tenth.
Problem 6
Find the length of XZ.
Problem 7
Find PS and SR.
Geometry – Chapter 7 – Notes and Examples
Section 5 – Using Proportional Relationships
__________________ ____________________________ is any method that uses formulas, similar figures, and/or
proportions to measure an object.
A ________ __________________ represents an object as smaller than or larger than its actual size.
The drawings _____________ is the ratio of the length in the drawing to the corresponding actual length.
For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual
distance.
Problem 1
A fire hydrant 2.5 feet high casts
a 5-foot shadow. How tall is a street light that
casts a 26-foot shadow at the same time?
x
2.5 ft
26 ft
5 ft
Problem 3
On a Wisconsin road map, Kristin measured a
distance of 11 in. from Madison to Wausau. The
scale of this map is 1 inch :13 miles. What is the
actual distance between Madison and Wausau to
the nearest mile?
Problem 2
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?
Problem 4
The rectangular central chamber of the Lincoln
Memorial is 74 ft long and 60 ft wide. What would
be the measurements of a scale drawing of the
floor of the chamber using a scale of 1 in : 20 ft?
Problem 5
Given that
∆LMN:∆QRS, find
the perimeter P
and area A of
∆QRS.
Problem 6
∆ABC ~ ∆DEF. Find
the perimeter and
area of ∆ABC.
Problem 7
Parallelogram PQRS ~ Parallelogram TUVW.
Find the perimeter and area of parallelogram
TUVW.
Problem 8
 EFG ~  HJK. Find the perimeter
and area of  HJK.