Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry congruent triangles? Then this Geometry lesson on Proving Triangles Congruent: SSS to HL explains each idea step by step, with worked examples.
Quick answer: Two triangles are congruent when rigid motions map one onto the other, so all corresponding sides and angles match. Triangles can be proved congruent with SSS, SAS, ASA, AAS or, for right triangles, HL. SSA and AAA do not work. After proving congruence, CPCTC says every remaining pair of corresponding parts is congruent.
Standards: G-CO.B.6, G-CO.B.7, G-CO.B.8, G-SRT.B.5 · Level: Grade 10 Geometry
What does congruent mean for triangles?
Two triangles are congruent when a sequence of rigid motions (translations, reflections and rotations) carries one exactly onto the other. Then all six corresponding parts, three sides and three angles, match. The order of letters in a congruence statement shows the matching: △ABC ≅ △DEF means ∠A ≅ ∠D, AB ≅ DE, and so on.
Which shortcuts prove that triangles are congruent?
You do not need all six parts. Three well-chosen parts are enough.
| Criterion | Parts that match | Note |
|---|---|---|
| SSS | Three sides | No angles needed |
| SAS | Two sides and the angle between them | Angle must be included |
| ASA | Two angles and the side between them | Side must be included |
| AAS | Two angles and a side not between them | Follows from ASA |
| HL | Hypotenuse and one leg of right triangles | Right triangles only |
Why does SSA fail?
Two sides and an angle that is not between them can build two different triangles. With ∠A = 30°, AB = 8 and BC = 5, side BC can swing to two spots on the ray from A, so AC may be about 3.9 or about 9.9. AAA also fails: it fixes shape but not size.
How do you write a congruence proof?
- Mark the Given facts on the figure, including shared sides and vertical angles.
- Pick a criterion: SSS, SAS, ASA, AAS or HL.
- List three congruent pairs, each with a reason.
- State that the triangles are congruent by that criterion.
- Use CPCTC (corresponding parts of congruent triangles are congruent) to reach the Prove statement.
Given: AB ≅ CB and AD ≅ CD in the kite above. Prove: ∠A ≅ ∠C. Since BD ≅ BD (reflexive), △ABD ≅ △CBD by SSS. By CPCTC, ∠A ≅ ∠C.
Do not call a pair of triangles congruent by SAS when the angle sits outside the two marked sides. The angle must be between them.
Name the triangles with vertices in matching order. A correct statement like △ABC ≅ △DEF tells you every corresponding pair at once.
- Congruent triangles have three pairs of congruent sides and three pairs of congruent angles.
- Valid criteria: SSS, SAS, ASA, AAS and HL for right triangles.
- SSA can describe two different triangles, so it does not prove congruence.
- CPCTC lets you conclude that remaining corresponding parts are congruent.
- A shared side is congruent to itself by the Reflexive Property.
Proving Triangles Congruent: SSS to HL: frequently asked questions
What does CPCTC stand for?
CPCTC means “corresponding parts of congruent triangles are congruent.” You use it only after you have proved the triangles congruent.
How do you choose between ASA and AAS?
Look at where the marked side sits. If it is between the two marked angles, use ASA. If it is next to only one of them, use AAS.
Why is HL allowed when SSA is not?
In a right triangle the angle opposite the hypotenuse is 90°, which removes the ambiguity. The remaining leg is forced by the Pythagorean Theorem.
Does AAA prove triangles congruent?
No. Triangles with equal angles have the same shape but can be different sizes, so they are similar and not necessarily congruent.
How do rigid motions define congruence?
If a translation, reflection, rotation or a sequence of them moves one triangle exactly onto another, the triangles are congruent. Rigid motions preserve every length and angle.
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