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 $$ 2 $$

Triangle Similarity Theorems Practice Problems & Examples

Master triangle similarity with AA, SAS, and SSS criteria through step-by-step practice problems. Learn to identify similar triangles and calculate similarity ratios.

📚Master Triangle Similarity Theorems with Interactive Practice

Apply AA similarity criterion using corresponding equal angles
Use SAS criterion with side ratios and included angles
Identify similar triangles using SSS side proportion method
Calculate similarity ratios between corresponding triangle sides
Prove triangle similarity using all three criteria methods
Solve real-world problems involving similar triangle applications

Understanding Similarity Theorems

Complete explanation with examples

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:

Conditions for the similarity between two triangles

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

Detailed explanation

Practice Similarity Theorems

Test your knowledge with 14 quizzes

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?

Examples with solutions for Similarity Theorems

Step-by-step solutions included

Exercise #1

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 #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 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 #5

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

Frequently Asked Questions

Everything you need to know about Similarity Theorems

What are the three triangle similarity criteria?

The three triangle similarity criteria are: 1) AA (Angle-Angle) - two triangles are similar if they have two equal corresponding angles, 2) SAS (Side-Angle-Side) - triangles are similar if two pairs of sides have equal ratios and the included angles are equal, and 3) SSS (Side-Side-Side) - triangles are similar if all three pairs of corresponding sides have equal ratios.

How do you prove triangles are similar using AA criterion?

To prove triangles are similar using AA criterion, you need to show that two pairs of corresponding angles are equal. Since the sum of angles in any triangle is 180°, if two angles are equal, the third angle must also be equal, making the triangles similar.

What is a similarity ratio in triangles?

A similarity ratio is the constant ratio between corresponding sides of similar triangles. For example, if triangle ABC is similar to triangle DEF with a ratio of 2:3, then each side of triangle ABC is 2/3 the length of the corresponding side in triangle DEF.

When do you use SAS similarity criterion?

Use SAS similarity criterion when you know the ratios of two pairs of corresponding sides are equal and the angles between these sides (included angles) are equal. This proves the triangles are similar without needing information about the third side or other angles.

How is SSS triangle similarity different from SSS congruence?

SSS similarity requires that the ratios of all three pairs of corresponding sides are equal (same proportion), while SSS congruence requires that all three pairs of corresponding sides are exactly equal in length. Similar triangles have the same shape but different sizes.

What's the easiest way to identify similar triangles?

The easiest method is often AA similarity - look for two pairs of equal corresponding angles. This frequently occurs with parallel lines creating equal angles, vertical angles, or shared angles between triangles.

Can you have similar triangles with different orientations?

Yes, similar triangles can have different orientations (rotated, flipped, or positioned differently). What matters is that corresponding angles are equal and corresponding sides have proportional lengths, regardless of how the triangles are positioned.

How do you find missing sides in similar triangles?

To find missing sides in similar triangles: 1) Identify the similarity ratio using known corresponding sides, 2) Set up a proportion equation with the known and unknown sides, 3) Cross-multiply and solve for the unknown side. Always ensure you're comparing corresponding sides.

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Triangle Similarity Theorems Practice Problems & Examples

Understanding Similarity Theorems

Conditions for the similarity between two triangles

Practice Similarity Theorems

Examples with solutions for Similarity Theorems

Frequently Asked Questions

What are the three triangle similarity criteria?

How do you prove triangles are similar using AA criterion?

What is a similarity ratio in triangles?

When do you use SAS similarity criterion?

How is SSS triangle similarity different from SSS congruence?

What's the easiest way to identify similar triangles?

Can you have similar triangles with different orientations?

How do you find missing sides in similar triangles?

Continue Your Math Journey

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