# Similarity Theorems - Examples, Exercises and Solutions

## Conditions for the similarity between two triangles

To demonstrate the similarity between triangles it is not necessary to show again and again the relationship between the three pairs of sides and the equivalence between all the corresponding angles. This would require too much unnecessary work.

There are three criteria by which we can see the similarity between triangles:

• Angle - Angle (AA): two triangles are similar if they have two equal angles.
• Side - Angle - Side (LAL): Two triangles are similar if the ratio between two pairs of sides and also the angle they form are equal.
• Side - Side - Side (LLL): Two triangles are similar if the ratio between all their sides (similarity ratio) is equal in both triangles.

### Suggested Topics to Practice in Advance

1. Similar Triangles

## Practice Similarity Theorems

### Exercise #1

Look at the following two triangles:

Angles B and D are equal.
Angles A and F are equal.

Which side corresponds to AB?

### Step-by-Step Solution

As we have two equal angles, we will use the angle-angle theorem to simulate triangles.

We will compare the vertices:$A=F,B=D$

According to the data it seems that:

Side AC corresponds to side EF.

Side BC corresponds to side DE.

Therefore, side AB corresponds to side FD.

$FD$

### Exercise #2

Look at the two triangles below:

Angle B is equal to angle E.
Angle A is equal to angle D.

Which angle corresponds to angle C?

### Step-by-Step Solution

As we have two pairs of corresponding angles, we will use the angle-angle theorem for triangle similarity.

Now that we know all angles are equal to each other, we note that the remaining angle that is equal and corresponds to angle C is angle F.

$F$

### Exercise #3

Angle B is equal to 40°

Angle C is equal to 60°

Angle E is equal to 40°

Angle F is equal to 60°

Are the triangles similar?

### Step-by-Step Solution

Given that the data shows that there are two pairs with equal angles:

$B=E=40$

$C=F=60$

The triangles are similar according to the angle-angle theorem, therefore triangle ABC is similar to triangle DEF.

Yes

### Exercise #4

Angle B is equal to 70 degrees

Angle C is equal to 35 degrees

Angle E is equal to 70 degrees

Angle F is equal to 35 degrees

Are the triangles similar?

### Step-by-Step Solution

The triangles are similar according to the angle-angle theorem.

Having two pairs of equal angles is sufficient to conclude that the triangles are similar.

Yes

### Exercise #5

Look at the two triangles below:

Angle B is equal to angle F.

Angle C is equal to angle D.

Which angle corresponds to angle A?

### Step-by-Step Solution

We use the angle-angle theorem to simulate triangles.

Let's observe the data we already have:

Angles B and F are equal.

Angle C is equal to angle D.

Therefore, the remaining angles must also be equal: angles A and E.

$E$

### Exercise #1

Look at the parallelogram ABCD below.

What can be said about triangles ACD and ABD?

### Step-by-Step Solution

According to the side-angle-side theorem, the triangles are similar and coincide with each other:

AC = BD (Any pair of opposite sides of a parallelogram are equal)

Angle C is equal to angle B.

AB = CD (Any pair of opposite sides of the parallelogram are equal)

Therefore, all of the answers are correct.

### Exercise #2

Are similar triangles necessarily congruent?

### Step-by-Step Solution

There are similar triangles that are not necessarily congruent, so this statement is not correct.

No

### Exercise #3

Are the triangles below similar?

### Step-by-Step Solution

The sides of the triangles are not equal and, therefore, the triangles are not similar.

No

### Exercise #4

In the image there are a pair of similar triangles and a triangle that is not similar to the others.

Determine which are similar and calculate their similarity ratio.

### Step-by-Step Solution

Triangle a and triangle b are similar according to the S.S.S (side side side) theorem

And the relationship between the sides is identical:

$\frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}$

$\frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3$

That is, the ratio between them is 1:3.

$a$ and $b$, similarity ratio of $3$

### Exercise #5

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

### Step-by-Step Solution

Triangle ABC is isosceles, therefore angle B is equal to angle C. We can calculate them since the sum of the angles of a triangle is 180:

$180-50=130$

$130:2=65$

As the triangles are similar, DE is parallel to BC

Angles B and D are corresponding and, therefore, are equal.

B=D=65

$65$°

### Exercise #1

Are the below triangles similar?

### Step-by-Step Solution

Use the similarity theorems.

Yes

### Exercise #2

Look at the following two triangles below:

Angles B and F are equal.

Angle C is equal to angle D.

Which side corresponds to AB?

### Video Solution

$EF$

### Exercise #3

Angle B is equal to 60°

Angle C is equal to 55°

Angle E is equal to 60°

Angle F is equal to 50°

Are these triangles similar?

No

### Exercise #4

Look at the two triangles below:

Angle B is equal to angle E.

Angle C is equal to angle F.

Which side corresponds to side AC?

### Video Solution

$DF$

### Exercise #5

Are the triangles below similar?