Congruent Triangles: Definition, 5 Rules, and Examples

In 7th grade math, students learn about congruent figures. Two triangles are congruent if they have the same size and shape, meaning all their sides and angles match—even if one triangle is flipped, turned, or slid on the plane.
Congruence is a foundational geometry concept that is often introduced during Pre-Algebra as students build spatial reasoning skills. Understanding congruent triangles helps students identify matching geometric figures, solve for unknown sides and angles, and write geometry proofs.

In this guide, we’ll explain what congruent means in math, introduce the five triangle congruence rules, and work through step-by-step examples.

What Does “Congruent” Mean?

In math, the word congruent means exactly the same in size and shape.

Two figures are congruent if one can be moved so that it fits perfectly over the other without being resized. The allowed movements are called rigid transformations:

• Translation: sliding a figure
• Rotation: turning a figure
• Reflection: flipping a figure

A congruent figure may face a different direction or appear in a different position, but its side lengths and angle measures remain unchanged. For example, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure.

What Are Congruent Triangles?

Congruent triangles have the same size and shape. Their corresponding sides and angles are equal.

This idea is often written as:

The symbol ≅ means “is congruent to.”

The order of the vertices is important. Writing △ABC ≅ △DEF communicates a different correspondence than writing △ABC ≅ △FED.

Corresponding Parts of Congruent Triangles

Once two triangles have been proven congruent, their corresponding sides and angles must also be congruent.

This principle is called CPCTC, which stands for:

Corresponding Parts of Congruent Triangles Are Congruent.

CPCTC is commonly used to find an unknown side length or angle measure after triangle congruence has already been established.

How Can You Tell If Two Triangles Are Congruent?

You do not need to compare all three sides and all three angles individually. In most problems, three carefully selected measurements are enough to prove that two triangles are congruent.

The following table provides a quick overview of the five triangle congruence rules.

Rule	Full Name	What Must Match
SSS	Side-Side-Side	All three corresponding sides
SAS	Side-Angle-Side	Two sides and the included angle
ASA	Angle-Side-Angle	Two angles and the included side
AAS	Angle-Angle-Side	Two angles and a non-included side
HL	Hypotenuse-Leg	The hypotenuse and one leg of right triangles

HL is also called RHS, or Right Angle-Hypotenuse-Side, in some math curricula.

Why AAA and SSA Do Not Prove Triangle Congruence

Not every combination of three measurements proves that two triangles are congruent.

AAA Does Not Prove Congruence

AAA stands for Angle-Angle-Angle.

Two triangles can have the same three angle measures but different side lengths. They will have the same shape but may have different sizes.

Therefore, AAA proves that triangles are similar, not necessarily congruent.

SSA Does Not Usually Prove Congruence

SSA stands for Side-Side-Angle, where the known angle is not between the two known sides.

The same SSA measurements can sometimes produce two different triangles. This is known as the ambiguous case.

Therefore, SSA does not generally prove triangle congruence.

How to Match Corresponding Parts of Congruent Triangles

Correctly matching corresponding vertices is essential when working with congruent triangles.

Consider: △ABC ≅ △PQR

The order tells us:

• A corresponds to P
• B corresponds to Q
• C corresponds to R

Therefore:

• AB corresponds to PQ
• BC corresponds to QR
• AC corresponds to PR
• ∠A corresponds to ∠P
• ∠B corresponds to ∠Q
• ∠C corresponds to ∠R

A common mistake is to match sides based only on where they appear in a diagram. Instead, use the order of the letters in the congruence statement.

Congruent Triangles vs. Similar Triangles

Congruent and similar triangles are related concepts, but they do not mean the same thing.

Feature	Congruent Triangles	Similar Triangles
Shape	Same	Same
Size	Same	May be different
Corresponding angles	Equal	Equal
Corresponding sides	Equal	Proportional
Symbol	≅	~
Resizing allowed	No	Yes

All congruent triangles are also similar because their corresponding sides have a scale factor of 1. However, similar triangles are not always congruent.

Example Problems: Congruent Triangles

Example 1: Find a Corresponding Angle

Problem: Given that △ABC ≅ △XYZ, m∠A = 40° and m∠B = 70°. What is the measure of ∠Z?

Solution:

Because △ABC ≅ △XYZ, the order of the letters shows that:

• A corresponds to X
• B corresponds to Y
• C corresponds to Z

Therefore: ∠C = ∠Z

First, use the triangle angle sum to find ∠C:

m∠C = 180° − (m∠A + m∠B)

m∠C = 180° − (40° + 70°)

m∠C = 70°

Since ∠C corresponds to ∠Z: m∠Z = 70°

Example 2: Find a Corresponding Side

Problem: Given that △PQR ≅ △MNO, PQ = 5 cm, QR = 7 cm, perimeter of △PQR = 20 cm. What is the length of side MO?

Solution:

From the congruence statement: △PQR ≅ △MNO

The corresponding vertices are:

• P corresponds to M
• Q corresponds to N
• R corresponds to O

Therefore, side PR corresponds to side MO.

First, find PR using the perimeter of △PQR:

PR = 20 cm − 5 cm − 7 cm

PR = 8 cm

Because corresponding sides of congruent triangles are equal: MO = PR = 8 cm

5 Common Mistakes With Congruent Triangles

Matching the Wrong Vertices

Do not assume vertices correspond based only on their positions in a picture. Use the order of the letters in the congruence statement.

Using AAA to Prove Congruence

AAA proves that two triangles have the same shape, but it does not prove that they have the same size.

Using SSA as a Congruence Rule

SSA does not usually determine a unique triangle, so it cannot generally prove congruence.

Using CPCTC Too Early

CPCTC can only be used after the triangles have been proven congruent through a valid rule such as SSS, SAS, ASA, AAS, or HL.

Confusing Congruent and Similar Triangles

Congruent triangles have equal corresponding sides. Similar triangles only require corresponding sides to be proportional.

More Congruent Triangles Practice Problems

Problem 1

Two triangles have corresponding side lengths of 4 cm, 6 cm, and 8 cm. Which congruence rule proves that they are congruent?

Answer: SSS

Problem 2

Two triangles have two corresponding sides equal and the angle between those sides equal. Which congruence rule applies?

Answer: SAS

Problem 3

Two right triangles have equal hypotenuses and one pair of equal legs. Which congruence rule applies?

Answer: HL

Problem 4

Two triangles have equal corresponding angles but side lengths of 3, 4, 5 and 6, 8, 10. Are they congruent?

Answer: No. The triangles are similar, but they are different sizes.

Problem 5

If △ABC ≅ △DEF and BC = 12 cm, what is the length of EF?

Answer: EF = 12 cm because BC corresponds to EF.

Summary: Congruent Triangles

Congruent triangles have exactly the same shape and size. Their corresponding sides and angles are equal, even when one triangle has been translated, rotated, or reflected.

Remember these five triangle congruence rules:

• SSS: three corresponding sides
• SAS: two sides and the included angle
• ASA: two angles and the included side
• AAS: two angles and a non-included side
• HL: the hypotenuse and one leg of right triangles

AAA proves similarity rather than congruence, and SSA does not usually provide enough information to prove congruence.

After two triangles have been proven congruent, use CPCTC to find unknown corresponding sides and angles.

Additional Math Topics for Triangles – with Free Worksheets

Looking for more geometry practice? Explore related topics below in these worksheets.

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More related articles

• Similar Triangles Explained: Geometry Rules & Examples

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