Topics
9. Dilations & Similarity
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United States of America
Grade 8

9.04 Triangles and angle angle (AA) similarity

Lesson
Practice

Introduction

We have learned that similar figures have congruent angles and sides that are proportional. Two figures are similar if one can be scaled up or down, and then rotated and/or reflected to match the other. We now understand that shapes that are congruent are also similar, but shapes that are similar are not always congruent.

We will now explore triangle similarity in more detail.

Triangles and angle-angle (AA) similarity

By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. We do not need to check all angles and sides in order to tell if two triangles are similar.

The angle-angle (AA) similarity criterion states that in two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.

In the diagram, \triangle GHI is similar to \triangle LMN:

Two similar triangles-triangle GHI with angle G equal to 28 degrees, angle I equal to 63 degrees and side length opposite to angle I measuring 29 units and triangle LMN with angle L equal to 28 degrees, angle N equal to 63 degrees and side length opposite to angle N measuring 29 units.

Since \angle I \cong \angle N and \angle G \cong \angle L, by AA similarity, \triangle GHI \sim \triangle LMN.

Examples

Example 1

Identify which two triangles are similar in the following set of triangles.

A
A triangle with two angles given: 72 degrees and 32 degrees and one side given with length of 27 units.
B
A triangle with two angles given: 73 degrees and 31 degrees and one side given with length of 17 units.
C
A triangle with two angles given: 73 degrees and 32 degrees and one side given with length of 27 units.
D
A triangle with two angles given: 73 degrees and 31 degrees and one side given with length of 7 units.
Worked Solution
Create a strategy

In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.

Apply the idea

For each triangle, think about what measurements are given and their positions in the triangle.

Do any of the other triangles have matching angles? Think about each option this way to help find the matching pair.

We can see that \triangle B has angles 31\degree and 73 \degree

A triangle with two angles given: 73 degrees and 31 degrees and one side given with length of 17 units.

which match the angles of \triangle D.

A triangle with two angles given: 73 degrees and 31 degrees and one side given with length of 7 units.

\triangle B \sim \triangle D by AA similarity criterion.

Example 2

Consider the following pair of similar triangles:

Two similar triangles-triangle GHI with angle G equal to 28 degrees, angle I equal to 63 degrees and side length opposite to angle I measuring 16 units and triangle LMN with angle L equal to x degrees, angle M equal to y degrees, side length opposite to angle N measuring 12 units and side length opposite to angle L measuring 9 units.
a

Find the corresponding angles.

Worked Solution
Create a strategy

A triangle is similar to another if one can be scaled up or down, and then rotated and/or reflected to match the other resulting to corresponding angles that are congruent.

Apply the idea

We can rotate \triangle LMN so that we can find the corresponding angles to \triangle GHI. Corresponding angles are congruent angles

Based on the figure, \angle G \cong \angle L, \angle H \cong \angle M and \angle N \cong \angle I.

b

Find the value of x.

Worked Solution
Create a strategy

Corresponding angles are congruent angles.

Apply the idea

We know that: \angle G \cong \angle L.

Measurement of \angle G = 28\degree

Measurement of \angle L=28 \degree

Threfore, x=28.

c

Find the value of y.

Worked Solution
Create a strategy

Corresponding angles are congruent angles.

The sum of the interior angles of a triangle is 180\degree.

Apply the idea

We know that: \angle M \cong \angle H.

To find the measurement of \angle H

\displaystyle m\angle G + m\angle I + m\angle H\displaystyle =\displaystyle 180\degreeThe sum of the interior angles of a triangle is 180\degree
\displaystyle 28\degree + 63\degree +m\angle H\displaystyle =\displaystyle 180\degreeSubstitute the values of \angle G and \angle H
\displaystyle m\angle H\displaystyle =\displaystyle 180\degree -28\degree - 63\degreeAddition property of equality
\displaystyle m\angle H\displaystyle =\displaystyle 89\degreeEvaluate

m\angle H \cong m\angle M, m\angle M = 89\degree; therefore, x=89

Idea summary

The angle-angle (AA) similarity criterion states that in two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.

Outcomes

8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

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