# How to Find Similar Figures?

Similar figures mean that two shapes are the same shape but have different sizes. In this guide, you will learn more about finding similar figures. Two shapes are called similar when both have the same properties but may not be the same. For example, the sun and the moon may seem the same size, but in reality, they are different sizes.

## Step by step guide tofinding similar figures

In geometry, when two shapes like triangle, polygon, quadrilateral, etc. have common dimensions or proportions but the size or length is different, they are considered similar figures. For example, two circles (of any radius) have the same shape but different sizes because they are similar. Look at the picture below.

The symbol for expressing similar figures is the same symbol for congruence i.e. “$$∼$$” but similarly does not mean the same in size. When the ratios of the corresponding sides are equal, the shapes are considered similar, that is, when dividing each set of corresponding side lengths, the number obtained is the scale factor. This number helps increase or decrease the size of the figures, but not in shape leaving them looking like similar figures.

### Similarity of triangles

Two triangles will be similar if the angles are equal (corresponding angles) and the sides are in equal proportion (corresponding sides). Similar triangles may have different lengths of sides of triangles, but their angles must be equal and their corresponding ratio of the lengths of the sides, or the scale factor must be the same.

If two triangles are similar that means,

• All corresponding angle pairs of triangles are equal.
• All corresponding sides of triangles are proportional.

#### $$AA$$ similarity criterion

The AA criterion for triangle similarity states that if the three angles of one triangle are equal to the three angles of the other triangle, respectively, then the two triangles will be similar.

Consider the following figure, in which $$ΔABC$$ and $$ΔDEF$$ are equiangular,i.e.,

• $$∠A = ∠D$$
• $$∠B = ∠E$$
• $$∠C = ∠F$$

We can say that these triangles are similar, using the $$AA$$ criterion.

#### $$SSS$$ similarity criterion

The $$SSS$$ similarity criterion states that if the three sides of a triangle are proportional to the other three sides of a triangle respectively, then the two triangles are similar. This means that any such pair of triangles will be equiangular(All corresponding angle pairs are equal) also.

Consider the following figure in which the sides of the two triangles $$ΔABC$$ and $$ΔDEF$$ are proportional, respectively:

That is, it is given that:

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$

#### $$SAS$$similarity criterion

The $$SAS$$ similarity criterion states that if two sides of a triangle are proportional to the two sides corresponding to another triangle, respectively, and if the included angles are equal, the two triangles are similar. If the equal angle is non-included, then the two triangles may not be similar. Consider the following figure:

It is given that:

$$\frac{AB}{DE}=\frac{AC}{DF}$$

$$∠A = ∠D$$

The $$SAS$$ criterion tells us that $$ΔABC ~ ΔDEF$$.

### Similarity of polygons

Similar polygons have the same shape but differ in size. There will be certain uniform ratios in similar polygons. In other words, the corresponding angles are congruent, but the corresponding sides are proportional.

There are two important properties of similar polygons:

• The corresponding angles are equal/congruent. (Both interior and exterior angles are the same)
• The ratio of the corresponding sides is the same for all sides. Hence, the perimeters are different.

Quadrilaterals are polygons that have four sides. The sum of the interior angles of a quadrilateral is $$360$$ degrees. Two quadrilaterals are similar when the three corresponding angles are the same and the two adjacent sides have equal ratios.

### Finding Similar Figures – Example 1:

Consider the following figure, and find the value of $$∠E$$.

Match the longest side with the longest side and the shortest side with the shortest side and check all three ratios:

• $$\frac{DE}{AB}=\frac{4.2}{6}=0.7$$
• $$\frac {DF}{AC}=\frac{2.8}{4}=0.7$$
• $$\frac{EF}{BC}=\frac{3.5}{5}=0.7$$

Thus, by $$SAS$$ similarity criterion, $$ΔABC ~ ΔDEF$$.

This means that they are also equiangular. Note that the equal angles will be as follows:

$$∠A = ∠D = 55.77°$$

$$∠C = ∠F = 82.82°$$

$$∠B = ∠E$$

Finally,

$$∠E = ∠B = 180° – (55.77° + 82.82°)$$

$$∠E = 41.41°$$

## Exercises for Finding Similar Figures

### Each pair of figures is similar. Find the missing side.

1. $$\color{blue}{100}$$
2. $$\color{blue}{9}$$
3. $$\color{blue}{12}$$

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