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Similar Triangles
Introduction: We examine triangles which are “similar” in appearance, the definition of similar triangles, and the ways we can use this information in measuring sides and angles.

Definition: Two triangles are said to be similar if they have the same angle measurements.

The Lesson:
The sides of two similar triangles do not have to be equal. However there is an important relationship among the sides of similar triangles: corresponding sides of similar triangles are in proportion. We illustrate these facts using the diagram below where we show two similar triangles ABC and QPR.

This relationship between these two triangles can be written as . Using this notation, we are saying that
These angles correspond to each other, and the naming of the triangles should put the angles A, B, and C in the same order as angles Q, P, and R. We can identify the corresponding sides in the same manner:
  • a corresponds to q, b corresponds to p, and c corresponds to r.
That the sides are in proportion gives us the following equation:

Equivalently, we could express these proportions using their reciprocals: br />

Let's Practice:
  1. Suppose . What is an equation that shows the proportionality of the corresponding sides?
Written in this order, we know that side a corresponds to x, side b corresponds to y and side c corresponds to side z. This gives us
  1. If and we know also that , what are the measures of angles Q, P, and R?
We use the correspondence of angles A, B, and C to angles Q, P, and R respectively. The corresponding angles of similar triangles have the same measure. Therefore we know that
Since these angles must have a sum of 180º,
Angle R has a measure of 53º
  1. If and sides a, b, and x are 3, 5, and 7 feet in measure respectively, what is the measure of side y?
Since the triangles are similar, we have the following proportion:
This gives us
3y = 35

Example If and , what are the measures of angles X, Y, and Z?
What is your answer?
Example and sides a, b, and q have measures of 5, 11, and 7 respectively, what is the measure of side p?
What is your answer?

M Ransom

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