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# Try some yourself

## Question 1

Which of these triangles are similar?

### Answer

Triangles a, c and g are similar since they have angles of 90°, 45° (and hence another angle of 45°).

Triangles b and f are similar since they have angles of 90°, 60° (and hence another angle of 30°).

Triangles d, e and h are similar since they have angles of 45° and 60° (and hence another angle of 75°).

## Question 2

An aluminium ladder can be used in three different ways:

The manufacturer says that in use, each segment of the ladder should make an angle of 20° with the vertical.

For each diagram, add construction lines and labels so as to identify two similar triangles. Are any of the similar triangles also congruent?

### Answer

There are many alternative solutions.

Here are some similar triangles which are identified by using the labels given in the diagram above:

##

domestic steps | ΔACD and ΔACB,
stair ladder | ΔSTR, ΔSTQ, ΔPQR, ΔSVU,
extension ladder | ΔFJH, ΔEIH, ΔEGF.

Some congruent triangles are

##

domestic steps | ΔACD and ΔACB,
stair ladder | ΔQST and ΔSTR.

## Question 3

This diagram shows the arrangement of the struts in a wall of a shed.

The lines OABC and DE are each horizontal. The struts EA and DC are parallel.

- (a) Which of these are right angles?

- (b) Write down two angles that are equal to

- (c) Several of the triangles formed by the struts are similar (that is, they are the same shape). Write down all the triangles that are similar to ΔOAF.

### Answer

3

- (a) All four of the given angles are right angles.

- (b) (which is the same as ) is equal to . They are alternate angles.

  (which is the same as ) is equal to . This is because ΔBCD and ΔOAF are similar: each has a right angle, and and are corresponding angles.

- (c) There are four triangles that are similar to ΔOAF: they are ΔOBD, ΔDEF, ΔOCD and ΔBCD.

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