Pairs of triangles are similar if they meet the conditions of any one of four standard tests.
Once proven to be similar, matching angles will be equal and matching sides will be proportional.
It is helpful if students are also familiar with the tests for congruence.
There are four similarity tests for triangles.
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Angle Angle Angle (AAA)
If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
It is sufficient to prove that only two pairs of angles are respectively equal to each other. Because the angle sum of a triangle is always 180°, the third pair of angles will automatically be equal. Re-stating this fact is not required when using the AAA test in a similarity proof.
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Side Angle Side (SAS)
If the ratio of the lengths of two sides of one triangle is equal to the ratio of the lengths of two sides of another triangle, and the included angles are equal, then the two triangles are similar.
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Side Side Side (SSS)
If each of the sides of one triangle can be matched up with each of the sides of another so that the ratios of matching sides are equal, then the two triangles are similar.
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Right-angle Hypotenuse Side (RHS)
If the ratio of the hypotenuse and one side of a right-angled triangle is equal to the ratio of the hypotenuse and one side of another right-angled triangle, then the two triangles are similar.
You can download further information about Similar triangles.
