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:

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

Which side corresponds to AB?

Video Solution

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 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.

Answer

FD FD

Exercise #2

Look at the two triangles below:

AAABBBCCCDDDEEEFFF

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

Which angle corresponds to angle C?

Video Solution

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.

Answer

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?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

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

B=E=40 B=E=40

C=F=60 C=F=60

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

Answer

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?

AAABBBCCCDDDEEEFFF

Video Solution

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.

Answer

Yes

Exercise #5

Look at the two triangles below:

AAABBBCCCDDDEEEFFF

Angle B is equal to angle F.

Angle C is equal to angle D.

Which angle corresponds to angle A?

Video Solution

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.

Answer

E E

Exercise #1

Look at the parallelogram ABCD below.

AAABBBDDDCCC

What can be said about triangles ACD and ABD?

Video Solution

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.

Answer

All answers are correct.

Exercise #2

Are similar triangles necessarily congruent?

Video Solution

Step-by-Step Solution

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

Answer

No

Exercise #3

Are the triangles below similar?

666999888555999888AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

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

Answer

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.

888444666999333666333111222AAABBBCCCGGGHHHIIIDDDEEEFFFABC

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:

GHDE=HIEF=GIDF \frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}

96=31=62=3 \frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3

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

Answer

a a and b b , similarity ratio of 3 3

Exercise #5

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

AAABBBCCCDDDEEE

Video Solution

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:

18050=130 180-50=130

130:2=65 130:2=65

As the triangles are similar, DE is parallel to BC

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

B=D=65

Answer

65 65 °

Exercise #1

Are the below triangles similar?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

Use the similarity theorems.

Answer

Yes

Exercise #2

Look at the following two triangles below:

AAABBBCCCDDDEEEFFF

Angles B and F are equal.

Angle C is equal to angle D.

Which side corresponds to AB?

Video Solution

Answer

EF 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?

AAABBBCCCDDDEEEFFF

Video Solution

Answer

No

Exercise #4

Look at the two triangles below:

AAABBBCCCDDDEEEFFF

Angle B is equal to angle E.

Angle C is equal to angle F.

Which side corresponds to side AC?

Video Solution

Answer

DF DF

Exercise #5

Are the triangles below similar?

666222444333111222AAABBBCCCDDDEEEFFF

Video Solution

Answer

Yes

Topics learned in later sections

  1. Similarity of Triangles and Polygons
  2. Similarity ratio
  3. Similarity of Geometric Figures