Similar triangles are triangles that have the same shape, but not necessarily the same size. They have corresponding angles that are equal and corresponding sides that are proportional.
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The Angle-Angle (AA) Similarity Criterion is one of the most commonly used methods to determine if two triangles are similar.
The ratio of the corresponding sides of similar triangles is called the scale factor, which is the same for all sides.
Similar triangles can be used to solve for unknown side lengths or angle measures in geometric applications.
The Pythagorean Theorem can be applied to solve for missing side lengths in right triangles, which are a special case of similar triangles.
Rectangles are a special case of similar triangles, where all angles are right angles and the ratio of the sides is the scale factor.
Review Questions
Explain how the Angle-Angle (AA) Similarity Criterion can be used to determine if two triangles are similar.
According to the Angle-Angle (AA) Similarity Criterion, if two triangles have two angles that are equal, then the triangles are similar. This means that the corresponding angles of the two triangles are equal, and the corresponding sides are proportional. The scale factor, which is the ratio of the corresponding sides, is the same for all sides of the similar triangles.
Describe how similar triangles can be used to solve for unknown side lengths or angle measures in geometric applications.
Similar triangles can be used to solve for unknown side lengths or angle measures in various geometric applications. By using the property that corresponding sides of similar triangles are proportional, you can set up a proportion to find the missing value. For example, if you know the lengths of two sides of one triangle and the corresponding sides of a similar triangle, you can use the proportion to solve for the length of the unknown side in the similar triangle. Similarly, if you know the angle measures of one triangle, you can use the property that corresponding angles are equal to determine the angle measures of the similar triangle.
Explain the relationship between similar triangles and the Pythagorean Theorem, and how this can be applied to solve for missing side lengths in right triangles.
The Pythagorean Theorem is a special case of the properties of similar triangles, as it applies to right triangles. In a right triangle, the two acute angles are always 90 degrees apart, making the triangle a special case of similar triangles. By applying the Pythagorean Theorem, you can solve for the missing side length in a right triangle if you know the lengths of the other two sides. This relationship between similar triangles and the Pythagorean Theorem allows you to use the properties of similar triangles to solve for unknown side lengths in a variety of geometric applications involving right triangles.
Related terms
Congruent Triangles: Congruent triangles are triangles that have the same size and shape, with all corresponding sides and angles being equal.
Angle-Angle (AA) Similarity Criterion: The Angle-Angle (AA) Similarity Criterion states that if two triangles have two angles that are equal, then the triangles are similar.