What is true about similar triangles

Table of Contents Show

Similar triangles
Corresponding angles
Corresponding sides
Naming similar triangles
Similar Triangles
Problem Solving

In this lesson we’ll look at the ratios of similar triangles to find out missing information about similar triangle pairs.

Similar triangles

In a pair of similar triangles, corresponding sides are proportional and all three angles are congruent. This means if you know two triangles are similar to one another you can use the information to solve for missing parts.

Corresponding angles

In a pair of similar triangles the corresponding angles are the angles with the same measure. In the diagram of similar triangles, the corresponding angles are the same color.

Example

If the two triangles in the diagram are similar, solve for the variable.

Corresponding sides touch the same two angle pairs.

Example

If ???\triangle XVY\sim \triangle XWZ???, solve for ???x???.

Year 10 Interactive Maths - Second Edition

Similar figures have the same shape (but not necessarily the same size) and the following properties:

Corresponding sides are proportional. That is, the ratios of the corresponding sides are equal.
Corresponding angles are equal.

For example, consider the following squares.

Thus the squares are similar figures as their corresponding sides are proportional and their corresponding angles are equal.

Note:

Each side of figure PQRS has been multiplied by 2 to obtain the sides of figure ABCD. The number 2 is called the scale factor.
Similar figures are equiangular (i.e. the corresponding angles of similar figures are equal).

Similar Triangles

Similar triangles can be applied to solve real world problems. For example, similar triangles can be used to find the height of a building, the width of a river, the height of a tree etc.

Recall that:

If two triangles are similar, then:

they are equiangular
the corresponding sides are in the same ratio
the angle included between any two sides of one triangle is equal to the angle included between the corresponding sides of the other triangle

Example 10

Find the value of x in the following pair of triangles.

Corresponding angles are marked in the same way in diagrams.

Example 11

Find the value of the pronumeral in the following diagram.

Solution:

Problem Solving

Example 12

Find the value of the height, h m, in the following diagram at which the tennis ball must be hit so that it will just pass over the net and land 6 metres away from the base of the net.

Solution:

So, the height at which the ball should be hit is 2.7 m.

Example 13

Adam looks in a mirror and sees the top of a building. His eyes are 1.25 m above ground level, as shown in the following diagram.

If Adam is 1.5 m from the mirror and 181.5 m from the base of the building, how high is the building?

Solution:

So, the height of the building is 150 m.

Note:

a. Equal angles are marked in the same way in diagrams.

b. Two triangles are similar if:

two pairs of corresponding sides are in the same ratio and the angle included between the sides is the same for both triangles.
the corresponding sides are in the same ratio.
the corresponding angles are the same.