ASA & AAS Triangle Congruence

SSS and SAS prove triangles congruent using mostly sides. ASA and AAS are the two criteria that lean on angles: each uses two pairs of congruent angles and one pair of congruent sides. The only difference between them is where that side sits.

In ASA the side is included — it connects the vertices of the two given angles. In AAS the side is somewhere else, opposite one of the given angles. Learn to check that one position and you will never mix the two up.

ASA: the side is between the angles

ASA stands for Angle-Side-Angle. If two pairs of corresponding angles are congruent and the pair of sides between those angles is also congruent, the triangles are congruent.

"Between" has a precise meaning: the side is included when its endpoints are the vertices of the two given angles. If the angles are at $J$ and $K$ , the included side is $\overline{JK}$ — no other side qualifies. In the figure below, both triangles have the two base angles marked and the side between them tick-marked — that is the ASA pattern.

AAS: the side is not between them

AAS stands for Angle-Angle-Side. Here the congruent side is not included — it is opposite one of the given angles instead of connecting them.

AAS works because of the Triangle Angle-Sum Theorem. Two angles of a triangle force the third: the three measures must total $180^\circ$ . So two pairs of congruent angles secretly give you all three pairs, and the given side becomes an included side between two known angles. Every AAS setup is an ASA setup in disguise.

Why AAA is not enough

Three pairs of congruent angles guarantee the same shape, not the same size. A small triangle and a large photocopy of it have identical angles but different side lengths — they are similar, not congruent.

Every valid congruence criterion includes at least one pair of congruent sides. That one side is what pins down the size.

Worked examples

Example 1: spotting ASA

In $\triangle PQR$ and $\triangle XYZ$ , $\angle P \cong \angle X$ , $\overline{PQ} \cong \overline{XY}$ , and $\angle Q \cong \angle Y$ . Which criterion applies?

List the given pairs in position order

\angle P,\; \overline{PQ},\; \angle Q

Check the side's endpoints —

\overline{PQ}

connects the vertices of the two given angles

P \text{ and } Q

The side is included, so this is Angle-Side-Angle

\text{ASA}

Answer: ASA

Example 2: spotting AAS

Two triangles each have angles of $41^\circ$ and $77^\circ$ , and in each triangle the side opposite the $41^\circ$ angle measures $15$ cm. Which criterion applies?

List the given pairs

\text{angle, angle, side}

Locate the side — it is opposite the

41^\circ

angle, so it does not connect the two given angles

\text{side not included}

Two angles and a non-included side is Angle-Angle-Side

\text{AAS}

Answer: AAS

Example 3: the side decides

In $\triangle ABC$ and $\triangle DEF$ , $\angle A \cong \angle D$ , $\angle B \cong \angle E$ , and $\overline{AC} \cong \overline{DF}$ . Which criterion applies?

The given angles sit at

A

and

B

, so the included side would be

\overline{AB}

\text{included side} = \overline{AB}

The congruent side is

\overline{AC}

instead — it touches only one of the given angles

\overline{AC} \neq \overline{AB}

Two angles and a non-included side

\text{AAS}

Answer: AAS

Try one yourself

Common questions

How do I tell whether the side is included?

Read the side's two endpoint letters. If both endpoints are vertices of the given angles, the side is included and the criterion is ASA. If either endpoint is not a given-angle vertex, the side is non-included and the criterion is AAS.

Is AAS the same as SAA?

Yes — the letters just read the parts in the opposite direction around the triangle. Most textbooks and answer keys write it AAS, so use that form.

Why does AAA fail when AAS works?

AAA gives no side at all, so the triangles can be different sizes — same shape, scaled up or down. AAS includes one pair of congruent sides, and that single side locks in the size.

Do I ever need all three angles?

No. Two pairs of congruent angles automatically force the third pair, because the angle measures of any triangle total $180^\circ$ . That is exactly why AAS is a valid criterion.

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ASA & AAS Triangle Congruence

ASA: the side is between the angles

AAS: the side is not between them

Why AAA is not enough

Worked examples

Example 1: spotting ASA

Example 2: spotting AAS

Example 3: the side decides

Try one yourself

Common questions

Keep learning

SSS & SAS Triangle Congruence

Congruent Figures & Corresponding Parts (CPCTC)

HL Congruence for Right Triangles

Triangle Congruence Proofs