What Are Similar Triangles? Definition, Properties, Formulas, Examples and Notes

Introduction

Similar triangles are triangles that have the same shape but do not necessarily have the same size. In similar triangles, the corresponding angles are equal and the corresponding sides are proportional.

Similarity is an important concept in geometry because it allows us to find unknown lengths and angles without measuring them directly.

For example, if two triangular shapes have corresponding angles equal and their corresponding sides are in the same ratio, the triangles are similar.

In this article, you will learn:

• The definition of similar triangles
• The difference between similar and congruent triangles
• Properties of similar triangles
• Conditions or criteria for proving triangles similar
• How to identify corresponding sides and angles
• The scale factor of similar triangles
• Step-by-step solved examples
• Applications of similar triangles in real life
• Common mistakes students should avoid
• Revision notes, MCQs, and a worksheet

What Are Similar Triangles?

Two triangles are called similar triangles if:

1. Their corresponding angles are equal.
2. Their corresponding sides are proportional.

The symbol for similarity is:$$\sim$$

For example, if triangle $ABC$ is similar to triangle $DEF$, we write:$$\triangle ABC \sim \triangle DEF$$

The order of the letters is important. It tells us which vertices correspond to each other:$$A \leftrightarrow D,\quad B \leftrightarrow E,\quad C \leftrightarrow F$$

Therefore:$$\angle A=\angle D$$$$\angle B=\angle E$$$$\angle C=\angle F$$

and$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$

Similar Triangles Definition

A simple similar triangles definition is:

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

Similar triangles have the same shape, although their sizes may be different.

Example

Suppose one triangle has sides:$$3,\ 4,\ 5$$

and another triangle has sides:$$6,\ 8,\ 10$$

Check the ratios:$$\frac{6}{3}=2$$$$\frac{8}{4}=2$$$$\frac{10}{5}=2$$

Since all corresponding sides have the same ratio, the triangles are similar.

The second triangle is an enlargement of the first triangle by a scale factor of $2$.

How to Understand Similar Triangles

Imagine two triangular road signs. One is small and another is twice as large, but both have exactly the same shape.

If every side of the larger triangle is twice the corresponding side of the smaller triangle, while the corresponding angles remain equal, the two triangles are similar.

Diagram Description

Imagine two triangles placed side by side:

• Triangle $ABC$ is the smaller triangle.
• Triangle $DEF$ is the larger triangle.
• $A$ corresponds to $D$.
• $B$ corresponds to $E$.
• $C$ corresponds to $F$.
• Each angle in the first triangle is equal to its corresponding angle in the second triangle.
• Every side of the second triangle is a fixed multiple of its corresponding side in the first triangle.

This represents:$$\triangle ABC\sim\triangle DEF$$

Properties of Similar Triangles

Similar triangles have several important properties.

1. Corresponding Angles Are Equal

If:$$\triangle ABC\sim\triangle DEF$$

then:$$\angle A=\angle D$$$$\angle B=\angle E$$$$\angle C=\angle F$$

2. Corresponding Sides Are Proportional

The corresponding sides have the same ratio:$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$

This common ratio is related to the scale factor.

3. Similar Triangles Have the Same Shape

Their sizes may be different, but their shapes are the same.

One triangle may be an enlargement or reduction of the other.

4. Their Perimeters Are in the Same Ratio as Corresponding Sides

If the scale factor is $k$, then:$$\frac{\text{Perimeter of larger triangle}} {\text{Perimeter of smaller triangle}} =k$$

5. Their Areas Are Related to the Square of the Scale Factor

If two similar triangles have a scale factor $k$, then:$$\frac{\text{Area of larger triangle}} {\text{Area of smaller triangle}} =k^2$$

For example, if the side lengths are in the ratio $2:3$, their areas are in the ratio:$$2^2:3^2=4:9$$

Corresponding Sides and Angles

Identifying corresponding parts is one of the most important steps when working with similar triangles.

Suppose:$$\triangle ABC\sim\triangle XYZ$$

Then:$$A\leftrightarrow X$$$$B\leftrightarrow Y$$$$C\leftrightarrow Z$$

Therefore, the corresponding sides are:

Triangle $ABC$	Triangle $XYZ$
$AB$	$XY$
$BC$	$YZ$
$AC$	$XZ$

Thus:$$\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$$

Always match the sides according to their corresponding angles.

Criteria for Similar Triangles

There are three commonly used criteria for proving that two triangles are similar.

1. AA Similarity Criterion

AA means Angle-Angle.

If two corresponding angles of one triangle are equal to two corresponding angles of another triangle, the triangles are similar.

For example:$$\angle A=\angle X$$

and$$\angle B=\angle Y$$

then:$$\triangle ABC\sim\triangle XYZ$$

The third pair of angles will also be equal because the sum of the angles of a triangle is $180^\circ$.

2. SSS Similarity Criterion

SSS means Side-Side-Side.

If the three corresponding sides of two triangles are proportional, the triangles are similar.

For example:$$\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$$

Therefore:$$\triangle ABC\sim\triangle XYZ$$

3. SAS Similarity Criterion

SAS means Side-Angle-Side.

If two corresponding sides are proportional and the included angle between them is equal, the triangles are similar.

For example:$$\frac{AB}{XY}=\frac{AC}{XZ}$$

and$$\angle A=\angle X$$

Therefore:$$\triangle ABC\sim\triangle XYZ$$

Similar Triangles vs Congruent Triangles

Students often confuse similar triangles with congruent triangles.

Similar Triangles	Congruent Triangles
Same shape	Same shape
Size may be different	Same size
Corresponding sides are proportional	Corresponding sides are equal
Corresponding angles are equal	Corresponding angles are equal
One can be an enlargement or reduction of the other	One can be superimposed exactly on the other
Symbol: $\sim$	Symbol: $\cong$

Important Point

Every pair of congruent triangles is also similar, because their corresponding sides are proportional with ratio $1$.

However, similar triangles do not have to be congruent.

For example, triangles with sides $3,4,5$ and $6,8,10$ are similar but not congruent.

Are Concurrent Triangles Similar?

The phrase “concurrent triangles” is not normally used as a criterion for triangle similarity.

The word concurrent generally describes lines, rays, or segments that meet at a common point. For example, three medians of a triangle are concurrent because they meet at the centroid.

Therefore, being concurrent does not by itself prove that two triangles are similar.

Students may sometimes confuse concurrent with congruent.

• Similar: same shape, corresponding sides proportional.
• Congruent: same shape and same size.
• Concurrent: lines or other geometric objects meet at one common point.

Similarity must be established using appropriate conditions such as AA, SSS, or SAS.

Scale Factor in Similar Triangles

The scale factor tells us how much larger or smaller one similar triangle is compared with another.

Suppose:$$\triangle ABC\sim\triangle DEF$$

and:$$AB=4\text{ cm}$$$$DE=8\text{ cm}$$

Then the scale factor from $ABC$ to $DEF$ is:$$k=\frac{8}{4}=2$$

Therefore, every corresponding side of triangle $DEF$ is twice the corresponding side of triangle $ABC$.

Formula for Finding an Unknown Side

For similar triangles:$$\frac{\text{Corresponding side}_1} {\text{Corresponding side}_2} = \frac{\text{Corresponding side}_3} {\text{Corresponding side}_4}$$

This proportion can be used to find an unknown length.

Worked Example 1: Finding an Unknown Side

Two similar triangles have corresponding sides of $4$ cm and $10$ cm. Another pair of corresponding sides is $6$ cm and $x$ cm. Find $x$.

Solution

Because the triangles are similar:$$\frac{4}{10}=\frac{6}{x}$$

Cross multiply:$$4x=60$$

Therefore:$$x=15$$

Answer

$$\boxed{x=15\text{ cm}}$$

Worked Example 2: Checking Whether Triangles Are Similar

Triangle $ABC$ has sides:$$5,\ 7,\ 9$$

Triangle $PQR$ has corresponding sides:$$10,\ 14,\ 18$$

Determine whether the triangles are similar.

Solution

Compare the corresponding sides:$$\frac{10}{5}=2$$$$\frac{14}{7}=2$$$$\frac{18}{9}=2$$

All three ratios are equal.

Therefore, the corresponding sides are proportional.

Answer

The triangles are similar by the SSS similarity criterion.

Worked Example 3: Using Equal Angles

Suppose:$$\angle A=50^\circ$$$$\angle B=70^\circ$$

in triangle $ABC$, and:$$\angle X=50^\circ$$$$\angle Y=70^\circ$$

in triangle $XYZ$.

Determine whether the triangles are similar.

Solution

We have:$$\angle A=\angle X$$

and:$$\angle B=\angle Y$$

Therefore, two corresponding angles are equal.

By the AA similarity criterion:$$\triangle ABC\sim\triangle XYZ$$

The third angles are also equal:$$\angle C=\angle Z=60^\circ$$

Answer

The triangles are similar by AA similarity.

Worked Example 4: Finding a Side Using Proportionality

Suppose:$$\triangle ABC\sim\triangle DEF$$

with:$$AB=8\text{ cm}$$$$DE=12\text{ cm}$$

and:$$BC=10\text{ cm}$$

Find $EF$.

Solution

Since the triangles are similar:$$\frac{AB}{DE}=\frac{BC}{EF}$$

Substitute the values:$$\frac{8}{12}=\frac{10}{EF}$$

Simplify:$$\frac{2}{3}=\frac{10}{EF}$$

Cross multiply:$$2EF=30$$

Therefore:$$EF=15$$

Answer

$$\boxed{EF=15\text{ cm}}$$

Worked Example 5: Similar Triangles and Area

Two similar triangles have corresponding sides in the ratio:$$2:5$$

Find the ratio of their areas.

Solution

For similar triangles, the ratio of areas is the square of the ratio of corresponding sides.

Therefore:$$\text{Area ratio}=2^2:5^2$$$$=4:25$$

Answer

$$\boxed{4:25}$$

Worked Example 6: Real-Life Height Measurement

A student wants to estimate the height of a tree. At the same time, a $2$-metre vertical pole casts a $3$-metre shadow. The tree casts a $12$-metre shadow.

Assuming the sun’s rays create similar triangles, find the height of the tree.

Solution

The triangles formed by the pole and tree are similar.

Let the height of the tree be $h$.

Set up the proportion:$$\frac{2}{3}=\frac{h}{12}$$

Cross multiply:$$3h=24$$

Therefore:$$h=8$$

Answer

The height of the tree is:$$\boxed{8\text{ m}}$$

Practical Applications of Similar Triangles

Similar triangles are useful in many real-life situations.

1. Measuring the Height of Tall Objects

The height of a tree, building, tower, or pole can sometimes be calculated by comparing its shadow with the shadow of an object of known height.

2. Surveying

Surveyors use geometric relationships, including triangle similarity, to determine distances and heights that may be difficult to measure directly.

3. Maps and Scale Drawings

Maps and diagrams often represent real objects at a smaller scale. The same proportional relationships are used when converting between drawing measurements and actual measurements.

4. Photography and Image Enlargement

When an image is enlarged without changing its shape, corresponding dimensions maintain a constant scale factor.

5. Engineering and Construction

Scaled drawings and models can use proportional relationships to represent larger structures accurately.

How to Solve Similar Triangle Problems

Follow these steps:

Step 1: Identify the Two Triangles

Determine which two triangles are being compared.

Step 2: Match Corresponding Angles

Use equal angles or markings in the diagram to identify corresponding vertices.

Step 3: Match Corresponding Sides

Once corresponding vertices are known, identify the corresponding sides.

Step 4: Select the Correct Similarity Criterion

Use:

• AA
• SSS
• SAS

when appropriate.

Step 5: Write a Proportion

For corresponding sides, write:$$\frac{a}{b}=\frac{c}{d}$$

Step 6: Solve for the Unknown

Use cross multiplication or another appropriate algebraic method.

Step 7: Check Your Answer

Make sure the answer follows the same scale relationship as the other corresponding sides.

Common Mistakes Students Make

Mistake 1: Matching the Wrong Sides

Students sometimes compare sides that are not corresponding.

Solution: Identify corresponding angles first, then match their opposite sides or connecting sides.

Mistake 2: Confusing Similar and Congruent

Similar triangles can have different sizes.

Congruent triangles must have the same size and shape.

Mistake 3: Reversing the Proportion

If:$$\frac{AB}{DE}$$

is used on one side of a proportion, the other pair should use the same order.

Correct:$$\frac{AB}{DE}=\frac{BC}{EF}$$

Do not mix:$$\frac{AB}{DE}=\frac{EF}{BC}$$

unless the proportion is deliberately rearranged correctly.

Mistake 4: Assuming Two Triangles Are Similar Because They Look Similar

A diagram is not always drawn to scale.

Similarity should be established mathematically using appropriate information.

Mistake 5: Confusing Concurrent With Congruent

Concurrent describes lines or other geometric objects meeting at one point. It does not mean that triangles have the same shape or size.

Frequently Searched Questions About Similar Triangles

What is the definition of similar triangles?

Similar triangles are triangles having equal corresponding angles and proportional corresponding sides.

What is the symbol for similar triangles?

The symbol for similarity is:$$\sim$$

For example:$$\triangle ABC\sim\triangle DEF$$

How do you know if two triangles are similar?

Two triangles can be shown to be similar using criteria such as AA, SSS, or SAS.

Are all congruent triangles similar?

Yes. Congruent triangles have equal corresponding sides, so their side ratio is $1$, making them similar as well.

Can similar triangles have different sizes?

Yes. Similar triangles have the same shape but can have different sizes.

What is the difference between similar and congruent triangles?

Similar triangles have proportional corresponding sides, whereas congruent triangles have equal corresponding sides as well as equal corresponding angles.

Are concurrent triangles always similar?

No. “Concurrent” is not a standard similarity criterion for triangles. Concurrency usually refers to lines or segments meeting at a common point.

Related Questions for Practice

Students studying this topic should also practise questions such as:

1. Find the missing side in two similar triangles.
2. Determine whether two triangles are similar using AA.
3. Determine whether two triangles are similar using SSS.
4. Determine whether two triangles are similar using SAS.
5. Find the scale factor between two similar triangles.
6. Find the perimeter of a similar triangle.
7. Find the area ratio of two similar triangles.
8. Use similar triangles to calculate the height of a tree.
9. Identify corresponding angles and sides.
10. Distinguish between similar and congruent triangles.

Short Conclusion

Similar triangles have the same shape, equal corresponding angles, and proportional corresponding sides. They can have different sizes because one triangle may be an enlargement or reduction of the other.

The three important similarity criteria are AA, SSS, and SAS. Once corresponding parts are identified correctly, proportions can be used to find unknown lengths, scale factors, and other quantities.

For students who want additional mathematics learning material, Nisar Math Academy provides mathematics notes, recorded video lectures, assessments, lesson plans, worksheets, solved questions, and other educational resources. Some learning resources are available free of charge, while full courses and additional resources are available through website membership.

Short Notes: Similar Triangles

Definition

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.$$\triangle ABC\sim\triangle DEF$$

Main Properties

• Same shape
• Corresponding angles are equal
• Corresponding sides are proportional
• Sizes may be different
• Perimeters have the same ratio as corresponding sides
• Areas have the square of the side ratio

Similarity Criteria

AA

Two corresponding angles are equal.

SSS

All three corresponding sides are proportional.

SAS

Two corresponding sides are proportional and the included angle is equal.

Important Formula

For similar triangles:$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$

Scale Factor

$$k=\frac{\text{corresponding side of second triangle}} {\text{corresponding side of first triangle}}$$

Area Ratio

If corresponding sides are in the ratio $a:b$, then:$$\text{Area ratio}=a^2:b^2$$

Similar vs Congruent

Similar: Same shape, possibly different size.

Congruent: Same shape and same size.

Concurrent: Lines or segments meeting at a common point; not a similarity criterion.

MCQs: Similar Triangles

1. What are similar triangles?

A. Triangles with only equal sides
B. Triangles with equal corresponding angles and proportional corresponding sides
C. Triangles with different shapes
D. Triangles with no equal angles

Correct Answer: B

2. What is the symbol for similarity?

A. $=$
B. $\cong$
C. $\sim$
D. $>$

Correct Answer: C

3. Which criterion uses two corresponding angles?

A. SSS
B. SAS
C. AA
D. RHS

Correct Answer: C

4. If two similar triangles have corresponding sides in the ratio $2:3$, their area ratio is:

A. $2:3$
B. $4:9$
C. $3:2$
D. $6:9$

Correct Answer: B

5. Triangle $ABC$ has sides $3,4,5$. A similar triangle has corresponding sides $6,8,10$. What is the scale factor from the first triangle to the second?

A. $1$
B. $2$
C. $3$
D. $4$

Correct Answer: B

6. If $\triangle ABC\sim\triangle XYZ$, which side corresponds to $AB$?

A. $YZ$
B. $XZ$
C. $XY$
D. $XZ$ and $YZ$

Correct Answer: C

7. If corresponding sides of two triangles are $5$ cm and $15$ cm, the scale factor is:

A. $2$
B. $3$
C. $5$
D. $10$

Correct Answer: B

8. Which of the following is NOT a standard triangle similarity criterion?

A. AA
B. SSS
C. SAS
D. AAA as a separately named criterion

Correct Answer: D

9. If two triangles have the same shape but different sizes, they may be:

A. Similar
B. Concurrent
C. Perpendicular
D. Parallel

Correct Answer: A

10. Two similar triangles have side ratio $3:4$. Their area ratio is:

A. $3:4$
B. $6:8$
C. $9:16$
D. $12:16$

Correct Answer: C

11. If two triangles have all three corresponding sides proportional, which criterion proves similarity?

A. AA
B. SSS
C. SAS
D. RHS

Correct Answer: B

12. If:

$$\frac{4}{8}=\frac{x}{12}$$

then $x$ is:

A. $4$
B. $6$
C. $8$
D. $10$

Correct Answer: B

13. Congruent triangles are:

A. Never similar
B. Always similar
C. Always different in shape
D. Similar only when their areas differ

Correct Answer: B

14. If the scale factor between two similar triangles is $3$, the ratio of their areas is:

A. $3$
B. $6$
C. $9$
D. $12$

Correct Answer: C

15. The term “concurrent” generally describes:

A. Equal triangles
B. Proportional sides
C. Lines or segments meeting at one point
D. Equal angles

Correct Answer: C

Worksheet / Assignment: Similar Triangles

Part A: Definitions and Concepts

1. Define similar triangles.
2. Write the symbol used to represent similarity between two triangles.
3. State any two properties of similar triangles.
4. Name the three commonly used criteria for proving triangles similar.
5. Explain the difference between similar and congruent triangles.

Part B: Identify Corresponding Parts

6. If:

$$\triangle ABC\sim\triangle PQR$$

write the corresponding vertices and corresponding sides.

7. If:

$$\triangle XYZ\sim\triangle LMN$$

write the proportion involving all three pairs of corresponding sides.

Part C: Numerical Problems

8. Two similar triangles have corresponding sides $6$ cm and $9$ cm. If another side of the smaller triangle is $8$ cm, find the corresponding side of the larger triangle.
9. Two similar triangles have corresponding sides in the ratio $3:5$. If a side of the smaller triangle is $12$ cm, find the corresponding side of the larger triangle.
10. Triangle $ABC$ has sides $4$, $6$, and $8$ cm. Another triangle has corresponding sides $8$, $12$, and $16$ cm. Determine whether the triangles are similar.
11. Two similar triangles have corresponding sides in the ratio $2:7$. Find their area ratio.
12. In two similar triangles:

$$\frac{5}{10}=\frac{x}{16}$$

Find $x$.

13. Two similar triangles have perimeters $18$ cm and $30$ cm. What is the ratio of their corresponding sides?

Part D: Word Problems

14. A $1.5$-metre pole casts a $2$-metre shadow. At the same time, a building casts a $20$-metre shadow. Assuming the triangles formed are similar, find the height of the building.
15. A small triangular model has a side of $8$ cm. Its corresponding side on a similar larger model is $20$ cm. If another side of the small model is $6$ cm, find the corresponding side of the larger model.

Answer Key

1. Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
2. $\sim$
3. Any two: corresponding angles are equal; corresponding sides are proportional; same shape; perimeter ratio equals side ratio.
4. AA, SSS, SAS.
5. Similar triangles have the same shape but may have different sizes; congruent triangles have the same shape and size.
6. $A\leftrightarrow P,\ B\leftrightarrow Q,\ C\leftrightarrow R$.
   $AB\leftrightarrow PQ,\ BC\leftrightarrow QR,\ AC\leftrightarrow PR$.

$$\frac{XY}{LM} = \frac{YZ}{MN} = \frac{XZ}{LN}$$

$$\frac{6}{9}=\frac{8}{x}$$$$x=12\text{ cm}$$

$$\frac{3}{5}=\frac{12}{x}$$$$x=20\text{ cm}$$

10. Yes. Each corresponding side of the second triangle is twice the corresponding side of the first.

$$2^2:7^2=4:49$$

$$\frac{5}{10}=\frac{x}{16}$$$$x=8$$

$$18:30=3:5$$

Therefore, the corresponding side ratio is:$$3:5$$

$$\frac{1.5}{2}=\frac{h}{20}$$$$2h=30$$$$h=15\text{ m}$$

$$\frac{8}{20}=\frac{6}{x}$$$$8x=120$$$$x=15\text{ cm}$$

FAQs About Similar Triangles

1. What are similar triangles?

Similar triangles have equal corresponding angles and proportional corresponding sides.

2. What is the definition of similar triangles in simple words?

They are triangles that have the same shape but may have different sizes.

3. What are the criteria for similar triangles?

The commonly used criteria are AA, SSS, and SAS.

4. Can two triangles be similar but not congruent?

Yes. For example, triangles with side lengths $3,4,5$ and $6,8,10$ are similar but not congruent.

5. How are corresponding sides identified?

First identify corresponding vertices using equal angles or diagram markings. The sides connecting corresponding vertices are corresponding sides.

6. What happens to the area when a similar triangle is enlarged?

If the side scale factor is $k$, the area scale factor is $k^2$.

7. Are concurrent triangles always similar?

No. Concurrency does not establish triangle similarity. Similarity requires conditions such as AA, SSS, or SAS.

8. Where are similar triangles used in real life?

They can be used for indirect measurements, surveying, scale drawings, construction, maps, and other applications involving proportional dimensions.

Final Revision Checklist

Before solving a problem involving similar triangles, remember:

• Identify the two triangles.
• Match corresponding vertices.
• Match corresponding sides.
• Check whether AA, SSS, or SAS applies.
• Write the proportion carefully.
• Keep corresponding quantities in the same order.
• Solve for the unknown.
• Check whether the answer follows the correct scale factor.
• Remember that similar triangles can have different sizes.
• Do not confuse similar, congruent, and concurrent.

Nisar Math Academy provides additional mathematics notes, recorded lectures, assessments, worksheets, lesson plans, and solved questions to support students in their mathematics learning.

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