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USA - Geometry Transformations, congruence, and proof

Triangle congruence (SAS, ASA, SSS, AAS, HL)

20 practice questions 2 video lessons Theory + worked examples

Triangle Congruence Criteria

California Geometry • Standard G-CO.8 • Transformations, Congruence & Proof

Triangle Congruence Criteria is a topic in Transformations, Congruence & Proof in the California Common Core State Standards. It is aligned to Standard G-CO.8, which requires students to explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions.

The triangle congruence criteria — SAS, ASA, SSS, AAS, and HL — prove two triangles congruent from limited information.

California Geometry › Transformations, Congruence & Proof › Triangle Congruence Criteria  —  Standard G-CO.8

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Practice questions

Every question with a fully worked solution.

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Watch 2 video(s)
  • Triangle Congruence Theorems Explained: ASA, AAS, HL Watch
  • Other triangle congruence postulates | Congruence | Geometry | Khan Academy Watch
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Theory

You do not need all six parts to prove two triangles congruent — a few matching parts are enough. The valid congruence criteria are:

  • SSS — three pairs of sides.
  • SAS — two sides and the included angle.
  • ASA — two angles and the included side.
  • AAS — two angles and a non-included side.
  • HL — hypotenuse and a leg, for right triangles only.
SSA and AAA are not valid. SSA can produce two different triangles, and AAA only guarantees similarity.
Side-Angle-Side (SAS) Two triangles are congruent if two sides and the angle between them match. SAS: two sides + included angle
SAS: two sides and the angle between them determine the triangle.
Triangle congruence criteria Triangle congruence criteria Triangle congruence criteria SSS • SAS • ASA AAS • HL (right triangles) not valid: SSA, AAA
The five valid criteria — and the two that fail.

The valid criteria:

\[\text{SSS},\ \text{SAS},\ \text{ASA},\ \text{AAS},\ \text{HL}\]
the valid triangle congruence criteria are SSS, SAS, ASA, AAS, and HL
“Included” is the key word for SAS (angle between the sides) and ASA (side between the angles).

How to prove triangles congruent

  1. Mark the given equal sides and angles on both triangles.
  2. Look for any shared side or angle, or a vertical-angle pair.
  3. Match the marks to a criterion (SSS, SAS, ASA, AAS, HL).
  4. State the congruence and, if needed, use CPCTC.
Example 1 — Choose the criterion
Two triangles have all three pairs of sides equal. Which criterion proves them congruent?
Solution

Three pairs of equal sides is SSS (Side-Side-Side).

the criterion is SSS
Example 2 — Included angle
Triangles share two sides and the angle between them. Name the criterion.
Solution

Two sides and the included angle is SAS.

the criterion is SAS
Example 3 — Right triangles
Two right triangles have equal hypotenuses and one equal leg. Are they congruent?
Solution

Yes, by HL (Hypotenuse-Leg), which applies only to right triangles.

yes, by the Hypotenuse-Leg criterion
Example 4 — Why SSA fails
Do two sides and a non-included angle (SSA) guarantee congruence?
Solution

No. SSA is not a valid criterion — the same two sides and non-included angle can form two different triangles (the ambiguous case). AAA also fails (it gives similar, not congruent, triangles).

no, SSA is not a valid congruence criterion

Common pitfalls

SSA is not valid. Two sides and a non-included angle can give two different triangles.
AAA gives similarity, not congruence. Same angles, but the sizes can differ.
Mind “included.” SAS needs the angle between the two sides; ASA needs the side between the two angles.

Frequently asked questions

What are the triangle congruence criteria?

SSS, SAS, ASA, AAS, and HL (for right triangles). Each proves two triangles congruent from a few matching parts.

What does the included angle mean in SAS?

The angle formed between the two given sides. SAS requires that specific angle, not just any angle.

Why is SSA not a valid criterion?

Two sides and a non-included angle can produce two different triangles (the ambiguous case), so it does not guarantee congruence.

What is the HL criterion?

Hypotenuse-Leg: two right triangles are congruent if their hypotenuses and one pair of legs are equal.

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