Similarity of Sides in Triangles

Similarity of Sides in Triangles

How to solve the similarity of sides in triangles problems: examples, and their solutions.

Example 1

Find the value of x. The measures of the left triangle's sides: 4, 5. The measures of the right triangle's sides: 6, x.

These two triangles have two congruent angles.
So, by the AA similarity,
these two triangles are similar.

Then their corresponding sides are proportional:
4 / 6 = 5 / x.

Example 2

Find the value of x. The measures of the line segments formed by two transversals: 7, 14, 3x + 2, and 16.

The red angles are vertical angles.
So they are congruent.

The red dot angles are
alternate interior angles in parallel lines.
So they are also congruent.

Then, by the AA similarity,
the purple triangle and the blue triangle are similar.

So their corresponding sides are proportional:
7 / 14 = (3x + 2) / 16.

Example 3

Find the value of x. The measures of the little right triangle's sides: 5, 3. The measures of the whole right triangle's height: 7.

From the given figure,
find two triangle that are congruent:
the little triangle and the whole triangle.

The red acute angles are the same angle.
And both triangle have the right angle.

So, by the AA similarity,
these two triangles are similar.

Then their corresponding sides are proportional:
3 / 7 = 5 / (5 + x).