Similar figures

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)

Similar triangles have:

• 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:

Corresponding sides given are the sides opposite to the angle marked with three arcs. i.e. in triangle A, side with 8 corresponds to side with 4 in B. So the ratio will be: $\frac{8}{4} = 2$

Now, we can find the length of the other sides by equating the value of the ratio we have obtained.

The side labeled a in A corresponds to side labeled 6 in B because they are both opposite to angles marked by two arcs.

Thus, $2 = \frac{a}{6}$

$12 = a$ (multiply by 6 both sides)

The side labeled 6 in A corresponds to the side labeled b in B because they are both opposite to the angle marked by one arc.

$$\text{Thus,} \quad 2 = \frac{6}{b}$$ $$2b = 6 \quad \text{(multiply by b both sides)}$$ $$b = 3 \quad \text{(divide by 2 both sides)}$$

Therefore, the length of sides a and b are 12 and 3 units respectively.

This is how we can use the property of similarity of triangles (i.e. corresponding sides have equal ratio of their lengths) to find the length of the missing sides.

Two triangles are similar if:

• All their angles are equal
• The corresponding sides are in the same ratio

But we don't need to know all three angles and all three sides, even two or three are enough.

The theory is also called side-splitter theorem. Let ABC be any triangle and DE is drawn parallel to BC, then AD/DB = AE/EC.

To show this is true draw a line DF parallel to EC:

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

Side AD corresponds to side DB and side AE corresponds to side DF, thus AD/DB = AE/DF but DF = EC, so AD/DB = AE/EC.

There are three ways to find that the two triangles are similar:

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 to $180^\circ$. In our case, our third angle will be: $180^\circ - (75^\circ + 40^\circ) = 65^\circ$

Therefore, AA can also be called AAA because when two angles are equal then all three angles must be equal.

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

For example:

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:

AB:XZ = 15:10 = 3:2 and BC:YZ = 21:14 = 3:2

Also, there is a matching angle of $75^\circ$ in between them.

The information is enough to tell us that the triangles are similar.

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:

a:x = 6:7.5 = 12:15 = 4:5 c:y = 4:5 b:z = 8:10 = 4:5

The ratios are all equal, so the triangles are similar.