# Similar Triangles

Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).

These triangles are all similar:

(Equal angles have been marked with the same number of arcs)

Notice that, as well as different sizes, some of them are turned or flipped.

For similar triangles:

All corresponding angles are equal

*and*

All corresponding sides have the same ratio

Also notice that the corresponding sides face the corresponding angles. For example the sides that face the angles with two arcs are corresponding.

## Corresponding Sides

In similar triangles, corresponding sides are always in the same ratio.

For example:

Triangles **R** and **S** are similar. The equal angles are marked with the same numbers of arcs.

What are the corresponding lengths?

- The lengths
**7**and**a**are corresponding (they face the angle marked with one arc) - The lengths
**8**and**6.4**are corresponding (they face the angle marked with two arcs) - The lengths
**6**and**b**are corresponding (they face the angle marked with three arcs)

## Calculating the Lengths of Corresponding Sides

We can sometimes calculate lengths we don't know yet.

- Step 1: Find
**the ratio**of corresponding sides - Step 2: Use that ratio to find the unknown lengths

### Example: Find lengths a and b of Triangle S

### Step 1: Find the ratio

We know all the sides in Triangle** R**, and

We know the side **6.4** in Triangle** S**

The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle **R**.

So we can match **6.4 with 8**, and so the ratio of sides in triangle** S** to triangle **R** is:

6.4 to 8

Now we know that the lengths of sides in triangle** S** are all **6.4/8 times** the lengths of sides in triangle **R**.

### Step 2: Use the ratio

**a** faces the angle with one arc as does the side of length 7 in triangle **R**.

**a**= (6.4/8) × 7 = 5.6

**b** faces the angle with three arcs as does the side of length 6 in triangle **R**.

**b**= (6.4/8) × 6 = 4.8

Done!

Did You Know? Similar triangles can help you estimate distances.