Explanation:
Triangles are similar when their corresponding angles are identical in measure, and their corresponding sides are proportional.
Problems involving similar triangles make use of these characteristics.*
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Triangles are named by naming their vertices, such as ΔABC, for example. If another triangle is similar, that fact is indicated using the wavy-line similarity symbol (~) between their names. For example:
ΔABC ~ ΔPQR
Corresponding angles are ones in the same position in the triangle's name. For example, angles B and Q correspond, because they are both named second in the similarity statement.
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Corresponding sides are named by using corresponding vertex names. For example, sides CA and RP are corresponding, because they are both named using the 3rd and 1st letters of the triangle names in the similarity statement.
Corresponding sides are proportional. This can be written different ways. Here are a couple of them:
AB : BC : CA = PQ : QR : RP
AB : PQ = BC : QR = CA : RP . . . . ratios are the same; they may be > 1 or < 1.
This means that if you know the ratio of any two corresponding sides in similar triangles, you know the ratio of all corresponding sides. This fact is used to solve problems involving side lengths.
Typically you might be given two sides of one triangle and two sides of a similar triangle (one corresponding). You can find the measures of the two missing sides by making use of the above relationships.
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* In general, problems in algebra and geometry make use of the relationships you learn. That is the point of learning the relationships. They let you write equations that relate what you know to what you don't know. Solving these equations lets you solve the problem.
