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. Let's Practice:
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:
That the sides are in proportion gives us the following equation:
- a corresponds to q, b corresponds to p, and c corresponds to r.
Equivalently, we could express these proportions using their reciprocals: br />
- 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
- 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
- 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