3.4: Using Similar Triangles

Angles of Similar Triangles

When two angles in one triangle are congruent to two angles in another triangle, the third angles are also congruent and the triangles are similar.

Indirect Measurement uses similar triangles to find a missing measure when it is difficult to find directly.

Identifying Similar Triangles

Tell whether the triangles are similar explain.

The triangles have two pairs of congruent angles.

So, the third angles are congruent, and the triangles are similar.

Identifying Similar Triangles

Write and solve an equation to find x.

Write and solve an equation to find x.

Tell whether the triangles are similar. Explain.

x 54 63180x 117 180

x 630

The triangles have two pairs of congruent angles. So, the third are congruent and the triangles are similar.

x 90 42 180x 132 180

x 480The triangles do not have two pairs of congruent angles. So, the triangles are not similar.

Practice

Solve the equation for x.

Tell whether the triangles are similar. Explain

Solve the equation for x.

x 2880 180x 108 180

x 720The triangles do not have two pairs of congruent angles. So, the triangles are not similar.

x 6690 180x 156 180

x 240

The triangles have two pairs of congruent angles. So, the third are congruent and the triangles are similar.

Using Indirect Measurement

a) <B and <E are right angles, so they are congruent. <ACB and <DCE are vertical angles, so they are congruent.

Because two angles in ABC are congruent to two angles in DEC, the third angles are also congruent and the triangles are similar.

b) The ratios of the corresponding side lengths in similar triangles are equal. Write and solve a proportion to find x.

So the distance across the river is 48 feet.

You plan to cross a river and want to know how far it is to the other side. You take measurements on your side of the river and make the drawing shown. (a) Explain why ABC and DEC are similar. (b) What is the distance x across the river?