Learn and apply the Angle-Angle-Angle and Angle-Angle criteria for proving triangle similarity.
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Geometric diagrams showing how equal angles lock in proportional sides
By definition, two triangles and are similar if all 6 conditions are met:
The Big Question: Do we really need to check all 6 conditions every single time to prove similarity?
Explanation of the Angle-Angle-Angle similarity criterion.
When you zoom in or scale a figure without distorting it, its shape stays intact and its angles remain unchanged.
The AAA (Angle-Angle-Angle) similarity criterion states that if the corresponding angles of two triangles are equal (, , ), then their corresponding sides are automatically in the same proportion:
Because equal angles force the side lengths to scale uniformly, the two triangles are guaranteed to be similar.
The mathematical statement for Angle-Angle-Angle similarity
Why two angles are enough.
To prove that two triangles are similar using angles, you might think you need to measure all three pairs of corresponding angles. This is the AAA (Angle-Angle-Angle) similarity criterion.
However, you can take a major shortcut called the AA similarity criterion. You only need to verify that two pairs of angles are equal!
Example 4 showing AA similarity using parallel lines and vertically opposite angles.
Problem. In the given figure, if line segment is parallel to line segment (), prove that .
Given:
Example 7: Find the length of a girl's shadow using similar triangles formed by a lamp-post.
A girl of height 90 cm (0.9 m) walks away from the base of a 3.6 m tall lamp-post at a speed of 1.2 m/s. After 4 seconds, the distance covered by the girl is . To find the length of her shadow , we set up a similarity relationship between and . Both and are equal to degrees because both the lamp-post and the girl stand vertically to the ground. Since is a common angle to both triangles, by the AA similarity criterion. Using the ratio of corresponding sides, we set up , which translates to . Solving yields a shadow length meters.