Triangle congruence: SSS, SAS, ASA, AAS and HL

The key idea

Congruent triangles are exact copies. One could be slid, turned or flipped to land perfectly on the other. The statement $\triangle ABC \cong \triangle DEF$ also tells you which parts match: the letters pair up in order.

\triangle ABC \cong \triangle DEF \;\Longrightarrow\; A \leftrightarrow D,\; B \leftrightarrow E,\; C \leftrightarrow F, \quad \overline{AB} \cong \overline{DE},\; \overline{BC} \cong \overline{EF},\; \overline{CA} \cong \overline{FD}

A triangle has six parts, three sides and three angles. Three well-chosen parts already lock its shape and size. These are the five shortcuts.

The five congruence shortcuts
Shortcut	What you need	Watch for
SSS	All three pairs of sides equal	Nothing extra needed
SAS	Two sides and the angle between them	The angle must sit where the two sides meet
ASA	Two angles and the side between them	The side must connect the two angles
AAS	Two angles and a side not between them	Works because the third angle is then fixed too
HL	Right triangles only: the hypotenuse and one leg	The right angle must be given or marked

Once two triangles are proven congruent, every other matching pair is equal as well. Proofs write this reason as CPCTC: corresponding parts of congruent triangles are congruent.

Worked examples

Example 1: name the shortcut

Problem In triangles PQR and XYZ, $PQ = XY = 7$ , $QR = YZ = 9$ , and $\angle Q = \angle Y = 50^\circ$ . Are the triangles congruent? If so, by which shortcut?

List what you know: two pairs of sides and one pair of angles.
Find where the two sides meet. PQ and QR share point Q. XY and YZ share point Y.
The given angles are at exactly those points, so each angle is between its two sides.
Two sides and the angle between them is SAS. Match the letters in order: P with X, Q with Y, R with Z.

Answer Yes, by SAS: $\triangle PQR \cong \triangle XYZ$ .

Example 2: use matching parts with algebra

Problem $\triangle ABC \cong \triangle DEF$ . $AB = 3x + 4$ , $DE = 5x - 10$ , and $BC = 2x + 1$ . Find x and the length of EF.

The order of the letters says AB matches DE, so they are equal.
$3x + 4 = 5x - 10$
Subtract $3x$ from both sides and add 10.
$14 = 2x$
Divide by 2.
$x = 7$
BC matches EF, so EF has the same length as BC.
$2(7) + 1 = 15$
Check: $AB = 3(7) + 4 = 25$ and $DE = 5(7) - 10 = 25$ . They match.

Answer $x = 7$ and $EF = 15$ .

Example 3: why SSA does not prove congruence

Problem A triangle has angle A = 30°, side AB = 10 and side BC = 6. Do these three facts give only one triangle?

Draw a 30° angle at A. Put point B on one ray so that $AB = 10$ . Point C has to land somewhere on the other ray.
Drop a perpendicular from B to the other ray. It makes a 30-60-90 triangle with hypotenuse 10, so the perpendicular is half of 10.
$\frac{1}{2} \cdot 10 = 5$
BC must be 6, which is longer than 5 but shorter than 10. A circle of radius 6 around B crosses the other ray in two places, one on each side of the perpendicular.
Call those points $C_1$ and $C_2$ . Triangles $ABC_1$ and $ABC_2$ both have angle A = 30°, AB = 10 and BC = 6.
But their third sides differ. With the law of cosines (or a careful drawing), $AC_1 \approx 5.34$ and $AC_2 \approx 11.98$ .

Answer No. Two different triangles fit, with AC about 5.34 or about 11.98, so SSA is not a congruence shortcut.

Example 4: a two-column proof

Segments AD and BC cross at point M, and M is the midpoint of both segments. Prove that $AB = DC$ .

Statement	Reason
1. M is the midpoint of AD and of BC	Given
2. $AM = DM$ and $BM = CM$	Definition of midpoint
3. $\angle AMB = \angle DMC$	Vertical angles are equal
4. $\triangle AMB \cong \triangle DMC$	SAS (steps 2 and 3, the angle at M is between the two sides)
5. $AB = DC$	CPCTC

This pattern comes up a lot. A midpoint hands you two pairs of equal sides, crossing segments hand you vertical angles, and SAS finishes the job.

Common mistakes

Matching the wrong letters. $\triangle ABC \cong \triangle DEF$ pairs A with D, not with F. Fix: match the equal angles first, then write both names in that order.
Calling SSA a shortcut. If the angle is not between the two sides, it does not prove congruence. The one exception is a right angle, which is HL.
Calling AAA congruence. Equal angles only prove the triangles are similar. One could be twice as big as the other.
Missing free information. A side shared by both triangles equals itself (the reflexive property), and crossing segments make vertical angles. Both count as given parts.
Using CPCTC too early. CPCTC only works after the triangles are proven congruent. It can never be the reason they are congruent.

Quick methods

Practice

5 practice questions

In $\triangle JKL$ and $\triangle RST$ , $JK = RS$ , $KL = ST$ , and $\angle K = \angle S$ . Which shortcut proves the triangles congruent?
1. SSS
2. SAS
3. ASA
4. Not enough information
Show answer
Answer: SAS
Angle K is where JK and KL meet, and angle S is where RS and ST meet, so each angle is between its two sides. That is SAS. SSS needs a third pair of sides, and ASA needs two pairs of angles.
Which set of matching parts does NOT prove two triangles congruent?
1. Two angles and the side between them
2. Three sides
3. Two sides and an angle that is not between them
4. Two angles and a side that is not between them
Show answer
Answer: Two sides and an angle that is not between them
That is SSA, which can fit two different triangles. The other three are ASA, SSS and AAS, which all prove congruence.
$\triangle ABC \cong \triangle PQR$ . Angle A measures 40° and angle Q measures 75°. What is the measure of angle R?
1. $40^\circ$
2. $65^\circ$
3. $75^\circ$
4. $105^\circ$
Show answer
Answer: $65^\circ$
Angle Q matches angle B, so B is 75°. Then angle C is $180 - 40 - 75 = 65$ , and angle R matches angle C. 105° is $180 - 75$ , which forgets angle A. 40° and 75° are the other two angles.
$\triangle DEF \cong \triangle XYZ$ , with $DE = 2x + 3$ and $XY = 4x - 9$ . What is DE?
1. $6$
2. $12$
3. $15$
4. $30$
Show answer
Answer: $15$
DE matches XY, so $2x + 3 = 4x - 9$ , which gives $x = 6$ . Then $DE = 2(6) + 3 = 15$ . 6 is x, not DE. 12 forgets the + 3, and 30 adds DE and XY together.
Triangles ABC and DEF have right angles at C and F. $AB = DE$ and $AC = DF$ . Which shortcut proves them congruent?
1. HL
2. SAS
3. ASA
4. Not enough information
Show answer
Answer: HL
AB and DE are across from the right angles, so they are the hypotenuses. AC and DF are legs. A hypotenuse and a leg in right triangles is HL. SAS would need angle A (between AB and AC), which is not given.

Frequently asked questions

Why doesn't SSA prove triangles congruent?

With two sides and an angle that is not between them, the third side can often swing into two different positions. Both positions make a real triangle with the same three given parts, but the triangles have different shapes. Example 3 above shows a case where the third side could be about 5.34 or about 11.98.

Is AAA a way to prove triangles congruent?

No. Three equal angles mean the triangles have the same shape, so they are similar. But one can be a blown-up copy of the other. To prove congruence you also need at least one pair of equal sides, which gives ASA or AAS.

What is the difference between ASA and AAS?

In ASA the known side is between the two known angles. In AAS it is not. Both prove congruence, because two angles fix the third angle, so AAS can always be turned into ASA. Teachers still want the right name, so check where the side sits.

What does CPCTC stand for?

Corresponding parts of congruent triangles are congruent. After you prove two triangles congruent, CPCTC lets you say any other matching pair of sides or angles is equal. It is usually the last line of a proof, never the reason the triangles are congruent.

Why does HL work when SSA does not?

HL is a special case of SSA where the known angle is 90°. A right angle leaves only one place for the third side, and the Pythagorean theorem gives its exact length from the hypotenuse and the leg. So only one triangle fits.

The key idea

Worked examples

Example 4: a two-column proof

Common mistakes

Quick methods

Practice

Frequently asked questions

Related

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