SSS & SAS Triangle Congruence

Two triangles are congruent when they are exact copies of each other — every side matches a side of the same length, and every angle matches an angle of the same measure. Written out, $\triangle ABC \cong \triangle DEF$ says the triangles match up in that letter order: $\angle A$ pairs with $\angle D$ , side $\overline{AB}$ pairs with $\overline{DE}$ , and so on.

Here's the good news: you never have to check all six pairs. Three well-chosen pieces are enough, and there are exactly four combinations that work — SSS, SAS, ASA, and AAS. Congruence problems are really just a matching game: list what's marked, figure out where the angle sits relative to the sides, and name the shortcut.

The four shortcuts

SSS (Side-Side-Side): all three pairs of corresponding sides are congruent. If the sides match, the angles have no choice but to match too — you can't build two differently-shaped triangles from the same three side lengths.

SAS (Side-Angle-Side): two pairs of sides are congruent, and the angles between those sides are congruent. The word between is the whole criterion — the angle must be the included angle, formed by the two given sides.

ASA (Angle-Side-Angle): two pairs of angles are congruent, and the sides between those angles are congruent. Again, position matters: the side must connect the two marked angles.

AAS (Angle-Angle-Side): two pairs of angles are congruent, and a pair of sides not between them is congruent. This works because once two angles match, the third angle matches automatically — so AAS is really ASA in disguise.

In the figure below, the tick marks show three pairs of congruent sides — one tick matches one tick, two ticks match two ticks, three ticks match three ticks. That's SSS, so the triangles are congruent.

Included vs. not included — the detail that decides everything

SAS and ASA both hinge on position. In SAS, the angle must sit between the two sides — it's the angle formed at the vertex where those two sides meet. In ASA, the side must connect the two angle vertices. If a marked angle or side is in the wrong position, the criterion changes or falls apart entirely.

A quick check: for SAS, look at the vertex of the marked angle and ask whether both marked sides touch it. For ASA vs. AAS, look at the marked side and ask whether both of its endpoints carry marked angles. Both endpoints marked means ASA; only one means AAS.

The two impostors: SSA and AAA

SSA — two sides and an angle that is not between them — does not prove congruence. With the same two sides and the same non-included angle, you can sometimes swing the third side into two different positions and get two genuinely different triangles. Test writers love disguising SSA as SAS, so always check where the angle sits before answering.

AAA — all three angles — doesn't work either, but for a different reason. Matching angles guarantee the same shape, not the same size: a small triangle and a poster-sized enlargement of it have identical angles. That situation is called similarity, not congruence. Every valid congruence shortcut includes at least one pair of sides.

Worked examples

Example 1: recognizing SSS

$\triangle ABC$ has sides $AB = 5$ , $BC = 7$ , and $AC = 9$ . $\triangle DEF$ has sides $DE = 5$ , $EF = 7$ , and $DF = 9$ . Are the triangles congruent?

Match the sides pair by pair

\overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}, \quad \overline{AC} \cong \overline{DF}

Count what's given: three pairs of sides, no angles needed

Three pairs of congruent sides is Side-Side-Side

\triangle ABC \cong \triangle DEF \text{ by SSS}

Answer: Yes — congruent by SSS

Example 2: recognizing SAS

Two triangles each have a side of $3$ , a side of $5$ , and a $40^\circ$ angle between those two sides. Which criterion proves them congruent?

List the given pairs in position order

\text{side, angle, side}

Check the angle's position: the

40^\circ

angle is formed by the

3

and

5

sides, so it is the included angle

Two sides and the included angle is Side-Angle-Side

\text{SAS}

Answer: SAS

Example 3: ASA or AAS?

Two triangles each have angles of $35^\circ$ and $82^\circ$ , and in each triangle the side opposite the $35^\circ$ angle measures $24$ . Which criterion applies?

List the given pairs

\text{angle, angle, side}

Check the side's position: it is opposite the

35^\circ

angle, so it does not connect the two marked angles — not included

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

\text{AAS}

Why it still works: the third angle is

180^\circ - 35^\circ - 82^\circ = 63^\circ

in both triangles, so an ASA setup is hiding inside

Answer: AAS

Example 4: the SSA trap

In two triangles, sides $\overline{AB}$ and $\overline{BC}$ are marked congruent to their partners, and $\angle A$ is marked congruent to its partner. Is this SAS?

Find the included angle for sides

\overline{AB}

and

\overline{BC}

: they meet at vertex

B

\text{included angle} = \angle B

The marked angle is

\angle A

, not

\angle B

— so the pattern is side-side-angle, not side-angle-side

SSA is not a valid congruence criterion

\text{no conclusion}

Answer: No — this is SSA, which proves nothing

Try one yourself

Common questions

How do I read the tick marks and arcs on a triangle diagram?

Tick marks pair up congruent sides: every side with one tick is congruent to every other side with one tick, two ticks match two ticks, and so on. Arcs do the same job for angles — matching numbers of arcs mean congruent angles. A small square marks a right angle.

Why isn't SSA a valid shortcut?

Because two sides and a non-included angle can sometimes build two different triangles — the third side can swing to two positions that both satisfy the given measurements. Since the information doesn't pin down one unique triangle, it can't guarantee congruence.

What is CPCTC and when do I use it?

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. It works in the direction opposite the shortcuts: first you prove two triangles congruent using SSS, SAS, ASA, or AAS, and then CPCTC lets you conclude that any remaining pair of sides or angles must match too.

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