If two triangles have a common angle and the ratios of the sides that include this angle are equal, then the triangles are similar by SAS similarity. That is, if all three corresponding ratios are the same, then the triangles are similar. This is another way to express SSS similarity. When interpreted for similar triangles, this shows that if you take the ratio of two sides in one triangle and the ratio of the two corresponding sides in the other, then these ratios will be the same.įor example, and the following ratios are all equal: Another proportion that gives the same cross-product is. In algebra, you can cross-multiply in a proportion such as, getting ad = bc. To write the similarity, we make sure the triangles are named with the smallest side as the first two letters and the largest with the 2 nd and 3 rd letters: Since these ratios are all equal, the triangles are similar by SSS similarity. To determine which sides "correspond," we list them from smallest to largest: That common ratio is either the scaling factor or the reciprocal of the scaling factor, depending on the direction in which we do the scaling.Įxample: Are these triangles similar? If so, write the similarity. This is what we call " SSS similarity." That is, if ratios of three pairs of corresponding sides of two triangles are equal, then the triangles are similar. We could form the reciprocals of the ratios, and they too will be the same: That is, the ratios of corresponding sides all reduce to the same fraction. If we form ratios of corresponding sides, we have: Notice that DE = 1.5 AB, EF = 1.5 BC, and DF = 1.5 AC. This includes triangles, and the scaling factor can be thought of as a ratio of side-lengths.įor example, triangle DEF is a scaled version of triangle ABC with a scaling factor of 1.5 (or 3/2), and we can write. As a consequence, their angles will be the same. Two geometric figures are similar if one is a scaled version of the other.
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