Determining Similarity with Angles

Lesson 1: Similar Polygons - Determining Similarity with Angles

   Constructing Knowledge

In general, for two polygons to be similar, their corresponding angles must be congruent (the same measure). However, congruent angle measures alone do not guarantee similarity. A combination of corresponding angle measure congruency and corresponding side length ratio equivalency is the only way to ensure similarity.

Consider the rectangles shown:


By definition, all four angles in a rectangle are right angles. However, it can be seen that the first pair of rectangles are not similar (even with congruent angle measures). In the second pair of rectangles, it appears possible that they are similar (congruent angle measures and same general shape), yet further investigation into the corresponding side length ratios is needed to be certain.

Consider the rhombi shown:


By definition, all four sides of a rhombus are equal in length. However, it can be seen that the rhombi are not similar because their angle measures are not congruent (despite having equal side lengths).

Because comparing corresponding angle measures is a little easier than comparing side length ratios, consider checking for similarity by first comparing corresponding angle measures.

If any of the corresponding angle measures are not the same, the polygons are not similar.

EXAMPLE 1

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Are the corresponding angles in the polygons congruent?

Solution

List all the corresponding angles

Relative position	Polygon ABCD	Corresponding angle on Polygon PQRS
	∠A = 52°	∠P = 52°
	∠B = 45°	∠Q= 45°
	∠C = 225°	∠R = 225°
	∠D = 38°	∠S = 38°

Because all corresponding angle measures in the polygon are the same, the polygons could be similar. However, to confirm similarity, corresponding side length ratios must be compared.




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