**TOPIC 6: SIMILARITY ~ MATHEMATICS FORM 2**

**Similarity**

**Similar Figures**

A corresponds to angle A’, angle B corresponds to B’, angle C corresponds to C’ also each pair of these corresponding sides bears the same ratio, that is:

Since all sides have the same ratio i.e. they are proportional and the corresponding angles are equal i.e. angle A = angle A’, angle B = angle B’ and angle C = angle C’, then the two figures are similar. The symbol for similarity is ‘**∼**‘

**Note**: all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand rectangles are not all similar to each other, isosceles triangles are not all similar to each other and ellipses are not all similar to each other.

Two

Triangles are similar if the only difference is size (and possibly the

need to Turn or Flip one around). The Triangles below are similar

(Equal Angles have been marked with the same number of Arcs)

- All their angles equal
- Corresponding sides have the same ratio

For example; Given similar triangles below, find the length of sides a and b

Solution

Since

we know that, similar triangles have equal ratio of corresponding

sides, finding the ratio of the given corresponding sides first thing:

**How to find if Triangles are Similar**

- All their equals are angles
- The corresponding sides are in the same ratio

**Intercept theorem**

The Triangles ADE and BD have exactly equal angles and so they are similar (recall that the two Triangles are similar by AA).

**1. AA (Angle-Angle)**:

this means, Triangles have two of their Angles equal. See an illustration below

If two of their Angles are equal then the third Angle must also be equal, because Angles of a Triangle add up to180^{0}. In our case, our third Angle will be:

^{0}– (75

^{0}+ 40

^{0}) = 65

^{0}

**2. SAS (Side-Angle-Side)**:

Means we have two Triangles where:

- The ratio between two sides is the same as the ratio between the other two sides
- The included Angles are equal

From our example, we see that, the side AB corresponds to side XZ and side BC corresponds to side YZ, thus the ratios will be:

^{0}in between them.

**3. SSS (Side-Side-Side)**:

Means we have three pairs of sides in the same ratio. Then the Triangles are Similar. For example;

In this example; the ratios of sides are:

Exercise 1

1. Use similarity to calculate side AB

2. ABC is a Triangle in which AC is produced to E and AB is Produced to D such that DE//BC. Show that AD:AB = DE:CB

5. Name two similar triangles in the figure below:

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