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 Practice Problems & Solutions Online

Master similar triangles with interactive practice problems. Learn AA, SAS, SSS criteria, similarity ratios, and area relationships through step-by-step solutions.

📚Master Similar Triangle Concepts Through Practice

Apply AA, SAS, and SSS similarity criteria to prove triangles are similar
Calculate similarity ratios between corresponding sides of similar triangles
Find missing side lengths using proportional relationships in similar triangles
Determine areas of similar triangles using the square of similarity ratios
Solve real-world problems involving similar triangles and scale factors
Identify corresponding angles and sides in similar triangle pairs

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

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?

Examples with solutions for Similar Triangles

Step-by-step solutions included

Exercise #1

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 #2

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 #3

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

Exercise #4

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 #5

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

Frequently Asked Questions

Everything you need to know about Similar Triangles

What are the three ways to prove triangles are similar?

The three similarity criteria are: 1) Angle-Angle (AA) - two pairs of corresponding angles are equal, 2) Side-Angle-Side (SAS) - two pairs of corresponding sides are proportional and the included angles are equal, 3) Side-Side-Side (SSS) - all three pairs of corresponding sides are proportional.

How do you find the similarity ratio of two triangles?

To find the similarity ratio, divide the length of any side in one triangle by the length of the corresponding side in the other triangle. All corresponding sides will have the same ratio in similar triangles.

What is the relationship between similarity ratio and area ratio?

The area ratio of similar triangles equals the square of the similarity ratio. For example, if the similarity ratio is 3:1, then the area ratio is 9:1.

How do you find missing sides in similar triangles?

Set up a proportion using the similarity ratio. If triangle ABC is similar to triangle DEF with ratio 2:1, and side AB = 6, then the corresponding side DE = 3. Use cross multiplication to solve for unknown sides.

What's the difference between similar and congruent triangles?

Similar triangles have the same shape but different sizes - their corresponding angles are equal and sides are proportional. Congruent triangles have identical shape AND size - all corresponding sides and angles are exactly equal.

Do similar triangles always have the same angles?

Yes, corresponding angles in similar triangles are always equal. This is a fundamental property of similarity - the triangles have the same shape, which means identical angle measures.

How do you use AA similarity criterion?

To use AA (Angle-Angle) similarity, show that two pairs of corresponding angles in the triangles are equal. Since the sum of angles in a triangle is 180°, the third pair of angles will automatically be equal too.

Can you have similar triangles with different orientations?

Yes, similar triangles can be rotated, flipped, or positioned differently. The key is that corresponding sides remain proportional and corresponding angles remain equal, regardless of orientation.

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