7.3. Similarity by AA http://www.ck12.org
Angles in Similar Triangles
The Third Angle Theorem states if two angles are congruent to two angles in another triangle, the third angles
are congruent too. Because a triangle has 180◦, the third angle in any triangle is 180◦minus the other two angle
measures. Let’s investigate what happens when two different triangles have the same angle measures. We will use
Investigation 4-4 (Constructing a Triangle using ASA) to help us with this.
Investigation 7-1: Constructing Similar Triangles
Tools Needed: pencil, paper, protractor, ruler
- Draw a 45◦angle. Extend the horizontal side and then draw a 60◦angle on the other side of this side. Extend
the other side of the 45◦angle and the 60◦angle so that they intersect to form a triangle. What is the measure
of the third angle? Measure the length of each side. - Repeat Step 1 and make the horizontal side between the 45◦and 60◦angle at least 1 inch longer than in Step
- This will make the entire triangle larger. Find the measure of the third angle and measure the length of each
side.
- This will make the entire triangle larger. Find the measure of the third angle and measure the length of each
- Find the ratio of the sides. Put the sides opposite the 45◦angles over each other, the sides opposite the 60◦
angles over each other, and the sides opposite the third angles over each other. What happens?
AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, the two
triangles are similar.
The AA Similarity Postulate is a shortcut for showing that twotrianglesare similar. If you know that two angles in
one triangle are congruent to two angles in another, which is now enough information to show that the two triangles
are similar. Then, you can use the similarity to find the lengths of the sides.
Example 1:Determine if the following two triangles are similar. If so, write the similarity statement.