Look at the parallelogram ABCD below. A A A B B B D D D C C C What can be said about triangles ACD and ABD?

They are equal in area and perimeter.

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. 8 8 8 4 4 4 6 6 6 9 9 9 3 3 3 6 6 6 3 3 3 1 1 1 2 2 2 A A A B B B C C C G G G H H H I I I D D D E E E F F F A B C

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

$$ a $$ and $$ c $$, similarity ratio of $$ 2 $$

$$ b $$ and $$ c $$, similarity ratio of $$ 1.5 $$

$$ b $$ and $$ c $$, similarity ratio of $$ 2 $$

Similar Triangles - Examples, Exercises and Solutions

Understanding Similar Triangles

Complete explanation with examples

What is triangle similarity?

Similar triangles are triangles for which there is a certain similarity ratio, that is, each of the sides of one triangle is in uniform proportion to the corresponding side in the other triangle. In addition, the angles at the same locations are also equal for the two similar triangles.

How do you prove the similarity of triangles?

To prove the similarity of triangles it is common to use one of three theorems:

Angle-angle (i.e., two pairs of equal angles in triangles).
Side-angle-side (similarity ratio of two pairs of sides in triangles and the angles trapped between them are equal)
Side-side-side (similarity ratio of three pairs of sides in triangles).

How do you prove the similarity of triangles

Similarities of triangles are expressed with the sign $∼$ .

Detailed explanation

Practice Similar Triangles

Test your knowledge with 9 quizzes

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?

Examples with solutions for Similar Triangles

Step-by-step solutions included

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.

Answer:

$FD$

Video Solution

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.

Answer:

$F$

Video Solution

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.

Answer:

Yes

Video Solution

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.

Answer:

Yes

Video Solution

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.

Answer:

$E$

Video Solution

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