Congruent Triangles

Learn what makes triangles congruent and how to prove congruence using SSS, SAS, ASA, AAS, and HL.

Intermediate25 minLesson
Printable worksheet

Definition

Two triangles are congruent if they have exactly the same shape and size. This means all three pairs of corresponding sides are equal, and all three pairs of corresponding angles are equal.
We write to show that triangle ABC is congruent to triangle DEF.
Key insight: You don't need to check all six measurements! There are five shortcut criteria to prove congruence:
CriterionWhat to Check
SSSThree pairs of sides
SASTwo sides and the included angle
ASATwo angles and the included side
AASTwo angles and a non-included side
HLHypotenuse and leg (right triangles only)

Try it now

Two triangles are congruent. What can you conclude?

Worked Examples

Prove that if cm, cm, and cm.

1

List the given information

cm, cm, cm → All three pairs of sides are given

2

Identify the congruence criterion

Three pairs of corresponding sides are equal → This matches SSS criterion

3

State the conclusion

By the SSS Congruence Postulate →

Common Mistakes

Using SSA (Side-Side-Angle) as a congruence criterion

Why it's wrong: SSA can produce two different triangles (the ambiguous case). The angle is not between the two sides, so it doesn't uniquely determine the triangle.

Correct: Only use SSS, SAS, ASA, AAS, or HL. Never use SSA (or 'ASS' - remember this by the inappropriate word it spells!).

Using AAA (Angle-Angle-Angle) to prove congruence

Why it's wrong: Having the same angles only guarantees similar triangles, not congruent ones. The triangles could be different sizes.

Correct: AAA proves similarity, not congruence. You need at least one pair of corresponding sides to prove congruence.

Confusing included and non-included angles/sides

Why it's wrong: SAS requires the angle to be BETWEEN the two sides. ASA requires the side to be BETWEEN the two angles.

Correct: Draw the triangle and label parts. For SAS: Side-ANGLE-Side (angle in middle). For ASA: Angle-SIDE-Angle (side in middle).

Forgetting to verify right angles for HL

Why it's wrong: The HL criterion only works for right triangles. Using it on non-right triangles is incorrect.

Correct: Always state that both triangles are right triangles before applying HL.

Interactive Visual

Triangle Explorer

By sides:Isosceles
By angles:Acute

Area:

20000.0 sq units

Drag vertices to create different triangle types.

Transformation Grid

T(3, 2)
-10-10-8-8-6-6-4-4-2-2224466881010xy(2, 1)(4, 1)(3, 4)(5, 3)(7, 3)(6, 6)
OriginalTransformed

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

15 problems
Problem 1 of 15
Easy

Two triangles are congruent. What can you conclude?

Why It Matters

Understanding congruent triangles is fundamental in geometry and has many practical applications:
  • Engineering: Trusses in bridges use congruent triangles for structural stability
  • Architecture: Symmetric designs rely on congruent shapes for balance and aesthetics
  • Manufacturing: Parts must be congruent to be interchangeable
  • Navigation: Triangulation uses congruent triangles to determine positions
Congruence is also the foundation for geometric proofs, which develop logical reasoning skills used throughout mathematics and beyond.

Real World Applications

Bridge Engineering

Truss bridges use congruent triangles to distribute weight evenly and provide structural stability.

Example:

The Warren truss design uses alternating congruent triangles. Each triangle has the same dimensions, ensuring equal load distribution.

1Try It Yourself

A bridge truss has triangles where each side measures 4 m, 4 m, and 5 m.

What criterion proves all triangles in this truss are congruent?

Step 1: Write the mathematical expression

Identify the criterion based on sides:

Manufacturing and Quality Control

Manufacturers must produce congruent parts that are interchangeable between products.

Example:

A bicycle company produces identical frame triangles. Each must be exactly 45 cm, 52 cm, and 58 cm to fit any bike of that model.

2Try It Yourself

Quality control checks two triangular brackets. Bracket A has sides 12 cm, 15 cm, 18 cm. Bracket B has sides 12 cm, 15 cm, 18 cm.

Can you conclude the brackets are congruent?

Step 1: Write the mathematical expression

Apply the appropriate criterion:

Land Surveying

Surveyors use triangulation to measure distances that cannot be measured directly.

Example:

To find the distance across a river, surveyors create a triangle on land that is congruent to one extending across the water.

3Try It Yourself

A surveyor creates a triangle with a 30 m base, angles of 48 degrees and 67 degrees at each end of the base.

What criterion determines this triangle uniquely?

Step 1: Write the mathematical expression

Identify: two angles and the side between them:

Key Takeaways

  • 1Congruent triangles have the same shape AND size - all corresponding sides and angles are equal
  • 2SSS: If all three pairs of sides are equal, triangles are congruent
  • 3SAS: If two sides and the INCLUDED angle are equal, triangles are congruent
  • 4ASA: If two angles and the INCLUDED side are equal, triangles are congruent
  • 5AAS: If two angles and a NON-included side are equal, triangles are congruent
  • 6HL: For RIGHT triangles only - if hypotenuse and one leg are equal, triangles are congruent
  • 7AAA only proves similarity, NOT congruence - triangles could be different sizes
  • 8SSA is NOT valid - it can produce two different triangles (ambiguous case)

Frequently Asked Questions

Congruent triangles have the same shape AND size - they are exact copies. Similar triangles have the same shape but can be different sizes - one is a scaled version of the other. Congruent triangles are always similar, but similar triangles are not always congruent.
Congruent triangles have the same shape AND size - they are exact copies. Similar triangles have the same shape but can be different sizes - one is a scaled version of the other. Congruent triangles are always similar, but similar triangles are not always congruent.
AAA (three equal angles) proves the triangles are similar, not congruent. You can have two triangles with identical angles but different sizes - like a small triangle and an enlarged photocopy of it. You need at least one side to fix the size.
Draw and label the triangles. Mark equal parts. Count: Do you have 3 sides (SSS)? 2 sides + included angle (SAS)? 2 angles + included side (ASA)? 2 angles + non-included side (AAS)? Right triangle with hypotenuse + leg (HL)?
Corresponding parts are in the same relative position in each triangle. If triangles ABC and DEF are congruent, then A corresponds to D, B to E, and C to F. Side AB corresponds to side DE, and so on.

Glossary

Congruent
Having exactly the same shape and size; all corresponding parts are equal
Corresponding parts
Parts (sides or angles) that are in the same position in two figures
Included angle
The angle formed between two sides (for SAS, the angle must be between the two given sides)
Included side
The side that connects two angles (for ASA, the side must be between the two given angles)
Hypotenuse
The longest side of a right triangle, opposite the right angle
Leg
Either of the two shorter sides of a right triangle that form the right angle
Postulate
A statement accepted as true without proof, used as a basis for reasoning
CPCTC
Corresponding Parts of Congruent Triangles are Congruent - used after proving congruence

More in This Topic

Introduction to Congruence

Intermediate25 min

Introduction to Similarity

Intermediate25 min

AA Similarity

Intermediate25 min

ASA and AAS Congruence

Intermediate25 min