- Two shapes are
**similar**if they have the same**shape**and their**corresponding sides**are in**proportion**- One shape is an
**enlargement**of the other

- One shape is an
- If two
**triangles**of different sizes have the**same angles**they are**similar**- Other shapes can have the same angles and
**not be similar**

- Other shapes can have the same angles and

** **

- To show that two non triangular shapes are similar you need to show that their corresponding sides are in proportion
- Divide the length of one side by the length of the same side on the other shape to find the
**scale factor**

- Divide the length of one side by the length of the same side on the other shape to find the
- If the scale factor is the same for all corresponding sides, then the shapes are similar
- If one shape can be shown to be an
**enlargement**of the other, then the two shapes are similar

- To show that two triangles are similar you simply need to show that their angles are the same
- This can be done through angle properties, look for isosceles triangles, vertically opposite angles and angles on parallel lines
- The triangles may not look similar and may be facing in different directions to each other, so concentrate on finding the angles
- it may help to sketch both triangles next to each other and facing the same direction

- If a question asks you to prove two triangles are similar, you will need to state that corresponding angles in similar triangles are the same and you will need to give a reason for each corresponding equal angle
- The triangles can often be opposite each other in an hourglass formation, look out for the vertically opposite, equal angles

- Proving two shapes are similar can require a lot of writing, you do not need to write in full sentences, but you must make sure you quote all of the
**keywords**to get the marks

a)

Prove that the two rectangles shown in the diagram below are similar.

Use the two lengths (15 cm and 6 cm) to find the scale factor.

Multiply this by the width of the smaller rectangle to see if it applies to the width as well.

**
The two rectangles are similar, with a scale factor of 2.5
**

b)

In the diagram below, *AB* and *CD* are parallel lines.

Show that triangles*ABX* and *CDX* are similar.

Show that triangles

State the equal angles by name, along with clear reasons.

Don’t forget to state that similar triangles need to have equal corresponding angles.

**
Angle AXB = angle CXD (Vertically opposite angles are equal) **

Angle ABC = angle BCD (Alternate angles on parallel lines are equal)

Angle BAD = angle ADC (Alternate angles on parallel lines are equal)

All three corresponding angles are equal, so the two triangles are similar

**How do I work with similar lengths?**- Equivalent
**lengths**in two similar shapes will be in the same ratio and are linked by a**scale factor**- Normally the first step is to find this scale factor
- STEP 1

Identify**equivalent**known lengths - STEP 2

Establish**direction**- If the scale factor is greater than 1 the shape is getting bigger
- If the scale factor is less than 1 the shape is getting smaller

- STEP 3

Find the**scale factor**- Second Length ÷ First Length

- STEP 4

Use**scale factor**to find the length you need

- STEP 4

- Equivalent

- If similar shapes overlap on the diagram (or are not clear) draw them separately
- For example, in this diagram the triangles ABC and APQ are similar:
- So we would redraw them separately before we start:

*ABCD *and *PQRS *are similar shapes.

Find the length of *PS.*

As the two shapes are mathematically similar, there will exist a value of *k *such that
and
.

*
*is known as the scale factor.

Form an equation using the two known corresponding sides of the triangle.

Solve to find

Substitute into

Solve to find .

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