Triangle Congruency- Rules and Practice Problems

What Triangle Congruency Actually Means

Two triangles are congruent if they have exactly the same three sides and exactly the same three angles. Not similar—congruent. Same size, same shape, one could stack perfectly on top of the other.

Geometry textbooks make this sound complicated. It's not. You just need to know which pieces of information guarantee a match.

The Five Rules That Actually Matter

1. SSS — Side-Side-Side

All three sides match. That's it. If triangle ABC has sides 3, 4, 5 and triangle DEF also has sides 3, 4, 5, they're congruent. No angles needed.

This works because three sides uniquely determine a triangle. There's only one triangle you can form with sides of lengths 3, 4, and 5.

2. SAS — Side-Angle-Side

Two sides and the included angle between them must match. The angle has to be sandwiched between the two sides—doesn't count if it's sticking out somewhere else.

So if AB = DE, AC = DF, and angle A = angle D, you've got congruence.

3. ASA — Angle-Side-Angle

Two angles and the side between them must match. Same deal as SAS—the side has to be the one connecting the two angles.

When you know two angles, you actually know all three (angles in a triangle add to 180°). So ASA is just as valid as SSS.

4. AAS — Angle-Angle-Side

Two angles and a side that isn't between them. This works because, again, two angles give you the third automatically. The side just needs to be somewhere on the triangle.

ASA and AAS are basically the same rule. Some textbooks list both, some combine them.

5. HL — Hypotenuse-Leg

This one is exclusive to right triangles. The hypotenuse and one leg must match between two right triangles.

Why does this work? Pythagorean theorem. If the hypotenuse and one leg match, the other leg must match too. It's a right triangle shortcut.

HL does NOT work for non-right triangles. Don't try it.

Rules Comparison Table

What Does NOT Work

SSA (Side-Side-Angle) is a trap. Two sides and an angle that isn't between them. This gives you ambiguous cases—sometimes one triangle, sometimes two, sometimes none. It's not a valid congruency rule. Teachers love putting this on tests to see if you're paying attention.

AAA (Angle-Angle-Angle) proves similarity, not congruency. You can have two different-sized triangles with identical angles. They're similar, not congruent.

How to Actually Prove Triangles Congruent

Step 1: Identify the triangles in question. Label them clearly—you'll need to match vertices.

Step 2: List what you know. Side lengths? Angle measures? Look for the given information in the problem.

Step 3: Check each rule in order. Does SSS work? SAS? Work through your options.

Step 4: State your conclusion. If you find a matching rule, write: "Triangle ABC ≅ Triangle DEF by [rule]."

The ≅ symbol means "is congruent to." Use it properly.

Practice Problems

Problem 1

Triangle ABC has sides AB = 5, BC = 7, AC = 9. Triangle DEF has sides DE = 5, EF = 9, DF = 7. Are they congruent?

Yes. SSS works—just match up the sides. AB = DE = 5, BC = DF = 7, AC = EF = 9. The order doesn't matter.

Problem 2

Triangle GHI has angle G = 40°, side GI = 6, angle I = 60°. Triangle JKL has angle J = 40°, side JL = 6, angle L = 60°. Congruent?

No. You have two angles and a side, but the side isn't between them. This is AAS, which is a valid rule—but only if you can confirm the side corresponds correctly. Here, the side JL is not between angles J and L. You'd need to verify the side is actually the included side or matches the corresponding non-included side. The problem doesn't establish this clearly, so you can't claim congruence.

Problem 3

Right triangle MNO has hypotenuse MO = 13 and leg NO = 5. Right triangle PQR has hypotenuse PR = 13 and leg QR = 5. Congruent?

Yes. HL works here. Both are right triangles, hypotenuses match (13), legs match (5). Done.

Problem 4

You're given that AB = CD, AD = CB, and diagonal AC bisects angle BAD and angle BCD. Prove triangle ABC ≅ triangle CDA.

Look at the given info. The diagonal AC is common to both triangles. AB = CD and AD = CB are given. The angle bisector info tells you angle BAC = angle DCA.

Now you have: AB = CD, AC = AC (reflexive), and angle BAC = angle DCA. That's SAS—two sides and the included angle between them. Triangles are congruent.

Common Mistakes That Cost You Points

• Using SSA and claiming it proves anything. It doesn't.
• Mixing up which side is the included side in SAS/ASA.
• Forgetting that HL only works for right triangles.
• Not using the reflexive property when a side or angle is shared between triangles.
• Writing the wrong rule name—SAS is not the same as ASA.

Quick Reference for Tests

When a congruency problem shows up, scan for these patterns:

• Three sides given → SSS
• Two sides and the angle between them → SAS
• Two angles with the side connecting them → ASA
• Two angles and any side → AAS
• Right triangle + hypotenuse + one leg → HL

If none of these patterns appear, you probably need more information before claiming congruence.