Triangle Congruence: Rules, Examples, Chart, and Practice

What is triangle congruence?

Triangle congruence means two triangles are exactly the same size and exactly the same shape.

If one triangle could be moved, turned, or flipped to sit perfectly on top of another triangle, the two triangles are congruent.

Congruent triangles have matching sides with equal lengths and matching angles with equal measures. Those matching parts are called corresponding parts.

Triangle congruence notation

A congruence statement must list the vertices in matching order.

For example, if △ABC ≅ △DEF, then A matches D, B matches E, and C matches F.

That order also tells you that side AB matches side DE, side BC matches side EF, and side AC matches side DF.

Main triangle congruence rules

A triangle has three sides and three angles, but you do not always need to know all six parts.

Triangle congruence rules tell you which smaller sets of facts are enough to prove that two triangles must match exactly.

The most common rules are SSS, SAS, ASA, AAS, and RHS/HL.

SSS: Side-Side-Side

Use SSS when all three sides of one triangle match all three sides of another triangle.

If the three side lengths match, there is only one possible triangle shape and size.

No angle measurements are needed for SSS because the three side lengths already force the angles.

SAS: Side-Angle-Side

Use SAS when two sides and the included angle match.

The included angle is the angle between the two known sides.

SAS works because the matching angle fixes how far apart the two sides open.

ASA and AAS

Use ASA when two angles and the side between them match.

Use AAS when two angles and a side not between them match.

AAS works because if two angles in a triangle are known, the third angle is forced by the angles in a triangle rule. For a focused example, review Angle-Angle-Side (AAS).

RHS or HL for right triangles

RHS means Right angle-Hypotenuse-Side. In some courses, the same rule is called HL, which means Hypotenuse-Leg.

This rule is only for right triangles.

If two right triangles have matching hypotenuses and one matching leg, the triangles are congruent. The right angle is the 90° angle, like the one explained in the right angle lesson.

Rules that are not enough

AAA is not enough for congruence. It proves the triangles have the same shape, but one triangle could still be a larger copy of the other.

SSA is usually not enough for congruence because it can sometimes make two different triangles.

When you are writing a proof, do not use AAA or SSA as triangle congruence reasons unless your course has given an extra special condition.

How to prove triangles congruent

Step 1: Mark the given equal sides and equal angles on the diagram.

Step 2: Match the corresponding vertices in the same order.

Step 3: Decide which rule fits the marked facts: SSS, SAS, ASA, AAS, or RHS/HL.

Step 4: Write the congruence statement, such as △ABC ≅ △DEF, with matching vertices in order.

Step 5: Use the rule name as the reason.

Worked example

Problem: Triangle ABC and triangle DEF have AB = DE, AC = DF, and angle A = angle D. Can you prove the triangles congruent?

Step 1: The two given sides meet at angle A in the first triangle.

Step 2: The matching two sides meet at angle D in the second triangle.

Step 3: The angle is included between the two matching sides.

Answer: The triangles are congruent by SAS.

CPCTC after congruence

After you prove two triangles congruent, you can use CPCTC.

CPCTC means Corresponding Parts of Congruent Triangles are Congruent.

In simple words: once the whole triangles match, every matching side and every matching angle also matches. This connects naturally to the idea of congruent angles.

Common mistakes

Do not write the vertices in a random order. The order must show which vertices correspond.

Do not use SAS unless the angle is between the two known sides.

Do not use ASA if the side is not between the two angles. That pattern is AAS.

Do not use AAA for congruence. AAA can show similarity, but it does not prove equal size.

Do not assume parts match just because the drawing looks close. Use marks, labels, measurements, or given statements.

Quick practice

1. Three sides match in two triangles. The rule is SSS.

2. Two sides and the angle between them match. The rule is SAS.

3. Two angles and the side between them match. The rule is ASA.

4. Two angles and a side not between them match. The rule is AAS.

5. Two right triangles have a matching hypotenuse and one matching leg. The rule is RHS or HL.

6. Three angles match, but no side length is given. That is AAA, so it is not enough for congruence.