Congruent Triangles Worksheet- Practice Problems and Answers

What Is Triangle Congruence?

Two triangles are congruent when they're identical in shape and size. Every corresponding side and angle matches exactly. It doesn't matter how the triangle is positioned or oriented—what matters is that all three sides and all three angles line up perfectly.

This is a foundational concept in geometry. It shows up constantly in proofs, construction, and real-world applications like engineering and architecture. If you can't identify congruent triangles quickly, you'll struggle with almost everything that comes after.

The Five Triangle Congruence Criteria

You don't need all six matching parts to prove triangles are congruent. Five shortcuts exist:

• SSS (Side-Side-Side) — All three sides match
• SAS (Side-Angle-Side) — Two sides and the included angle match
• ASA (Angle-Side-Angle) — Two angles and the included side match
• AAS (Angle-Angle-Side) — Two angles and a non-included side match
• HL (Hypotenuse-Leg) — Right triangles only: hypotenuse and one leg match

Important: SSA (two sides and a non-included angle) does NOT guarantee congruence. It's the trap question that trips up students constantly. Don't use it.

Quick Comparison Table

How To Identify Which Criterion to Use

Look at what information you're given and match it to a criterion:

1. Scan the diagram for marked sides (hash marks, double marks, etc.)
2. Scan for marked angles
3. Count how many matching parts you have
4. Check if the angle is included between two sides, or if the side is between two angles
5. Apply the matching criterion

Example: If you see two sides with a angle mark between them, that's SAS. If you see two angles with a side mark between them, that's ASA or AAS (depends on side placement).

Practice Problems with Answers

Problem 1

In the diagram, AB = DE, BC = EF, and AC = DF. Which criterion proves △ABC ≅ △DEF?

Answer: SSS. All three corresponding sides are equal.

Problem 2

Given: ∠A = ∠D, AB = DE, ∠B = ∠E. Which criterion proves the triangles congruent?

Answer: ASA. Two angles with the side between them.

Problem 3

△XYZ and △PQR are right triangles. XZ = PR (hypotenuse) and XY = PQ (leg). Which criterion applies?

Answer: HL. Right triangle with hypotenuse and one leg matching.

Problem 4

Given: ∠M = ∠S, ∠N = ∠T, and MN = ST. Can you prove congruence? Which criterion?

Answer: Yes. AAS. Two angles and a side that is NOT between them.

Problem 5

Two triangles have sides 5cm, 7cm, and 9cm. Are they necessarily congruent?

Answer: Yes. SSS applies. If all three sides match, the triangles are congruent. This is the SSS criterion.

Problem 6

You're given: side AB = side AD, side BC = side DC, and diagonal AC is shared. Which triangles are congruent?

Answer: △ABC ≅ △ADC by SSS. All three sides correspond: AB = AD, BC = DC, and AC = AC (common side).

Common Mistakes to Avoid

• Using SSA — This doesn't work. Two sides and an angle that isn't included between them is not a valid criterion.
• Mixing up ASA and AAS — ASA has the side between the angles. AAS has the side outside the angles. Both work, but they're not interchangeable.
• Forgetting HL restrictions — HL only works for right triangles. Don't apply it to acute or obtuse triangles.
• Assuming reflection counts differently — A flipped or rotated congruent triangle is still congruent. Orientation doesn't matter.

Getting Started: How To Solve Any Triangle Congruence Problem

Follow this step-by-step process:

1. Label the triangles — Give each vertex a clear letter. This prevents confusion later.
2. Mark the given information — Use hash marks for sides, arcs for angles. Visual learners need this.
3. Identify matching parts — Side-to-side, angle-to-angle. Write down what you know.
4. Count the matching parts — Do you have three sides? Two sides and an angle? Two angles and a side?
5. Apply the correct criterion — Match your count to SSS, SAS, ASA, AAS, or HL.
6. Write the congruence statement — Example: △ABC ≅ △DEF. The order matters. Match vertices in the correct sequence.

That's it. No fancy tricks. Just identify, match, and state.

Why This Matters Beyond the Worksheet

Triangle congruence isn't abstract busywork. It's how engineers ensure load-bearing structures stay stable. It's how architects calculate roof pitch. It's how GPS systems triangulate your position.

Master the criteria now, and proofs, constructions, and coordinate geometry become infinitely easier. Drag your feet, and you'll be re-learning this in every subsequent math class.