Similarity in Triangles

Lesson 1: Similar Polygons - Similarity in Triangles

   Constructing Knowledge

You may recall learning some special properties about triangles. For example, the interior angle measures in a triangle add up to 180° and if two angles in a triangle are known, the third can easily be calculated by subtracting the two known angles from 180°. When comparing triangles for similarity, comparing angle measures alone is enough. If all the corresponding angle measures are the same, the triangles are similar.

Typically when determining similarity in polygons, corresponding side length ratios must also be compared. However, with triangles, this is not required.

In triangles JKL and MNO, all corresponding angle measures are equal, so the triangles are similar.

In right angle triangles, if one pair of corresponding acute angle measures is the same, the triangles are similar.



For example, triangles ABC and EDF both have a 27° angle and a right angle. The third angle in these triangles is 63°, which is calculated as follows:

∠A and ∠E = 180° − 90° − 27°
∠A and ∠E = 63°

Triangles ABC and DEF are similar.

EXAMPLE 1

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Which of the following pairs of triangles are similar?

Triangles	Are they Similar?

Solution

Triangles	Are they Similar?
	∠X = 180° − 90° − 40° = 50° Because angle X is 50° and one of the acute angles in triangle RST is 50°, these triangles are similar.
	∠A = 180° − 90° −75° = 15° Neither acute angle in triangle TUV is 15°, which means these triangles are not similar.
	∠I = 180° − 120° − 30° = 30° ∠Q = 180° − 30° −20° = 130° The interior angles for each of the triangles are: Triangle GHI: 120°, 30°, 30° Triangle PQR: 130°, 30°, 20° The triangles are not similar because they each contain different angle measures.

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