Congruent Triangles

Two triangles are congruent when they have exactly the same shape and the same size. Every side and every angle in one triangle matches a corresponding side or angle in the other. You can think of congruent triangles as perfect copies — if you could pick one up and place it on top of the other (possibly flipping or rotating it), they would match exactly.

We write $\triangle ABC \cong \triangle DEF$ to mean that the vertices correspond in order: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$. This order matters — it tells you which parts are equal.

Why Congruence Matters

Proving that two triangles are congruent is one of the most useful tools in geometry. Once you know two triangles are congruent, you immediately know that all six pairs of corresponding parts (three sides and three angles) are equal — even the ones you have not measured yet. This is the basis for proving that segments are equal, angles are equal, or lines are parallel throughout geometry.

The Five Congruence Criteria

You do not need to check all six parts. Any one of the following five criteria is enough to guarantee that two triangles are congruent.

SSS (Side-Side-Side)

If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.

$\overline{AB} = \overline{DE}, \quad \overline{BC} = \overline{EF}, \quad \overline{AC} = \overline{DF} \implies \triangle ABC \cong \triangle DEF$

SSS works because three fixed side lengths determine a triangle’s shape completely — there is no way to change the angles without changing a side.

SAS (Side-Angle-Side)

If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.

$\overline{AB} = \overline{DE}, \quad \angle B = \angle E, \quad \overline{BC} = \overline{EF} \implies \triangle ABC \cong \triangle DEF$

The word “included” is critical. The angle must be the one formed between the two known sides.

ASA (Angle-Side-Angle)

If two angles and the included side (the side between those two angles) of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.

$\angle A = \angle D, \quad \overline{AB} = \overline{DE}, \quad \angle B = \angle E \implies \triangle ABC \cong \triangle DEF$

AAS (Angle-Angle-Side)

If two angles and a non-included side (a side that is NOT between the two known angles) of one triangle are equal to the corresponding parts of another, the triangles are congruent.

$\angle A = \angle D, \quad \angle B = \angle E, \quad \overline{BC} = \overline{EF} \implies \triangle ABC \cong \triangle DEF$

AAS works because if two angles match, the third angle is determined (they must sum to $180\degree$), so you effectively know all three angles plus a side — which is enough.

HL (Hypotenuse-Leg) — Right Triangles Only

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, the triangles are congruent.

$\text{hypotenuse}_1 = \text{hypotenuse}_2, \quad \text{leg}_1 = \text{leg}_2 \implies \triangle_1 \cong \triangle_2$

HL only applies to right triangles. It works because the Pythagorean theorem lets you find the third side from the hypotenuse and one leg, reducing this to SSS.

Why SSA Does NOT Work

You might wonder: if SAS works, why not SSA (Side-Side-Angle) where the angle is not between the two sides?

SSA is called the ambiguous case because two sides and a non-included angle can sometimes produce two different triangles. Imagine you know sides $a$ and $b$ and the angle opposite side $a$. Depending on the lengths, the second side might “swing” to two different positions, creating two distinct triangles that both satisfy the given conditions.

Because SSA does not guarantee a unique triangle, it is not a valid congruence criterion. This is the most common mistake students make when identifying congruence criteria.

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

CPCTC stands for “Corresponding Parts of Congruent Triangles are Congruent.” It is not a way to prove triangles are congruent — it is what you use after you have already proven congruence.

The logic works like this:

1. Prove $\triangle ABC \cong \triangle DEF$ (using SSS, SAS, ASA, AAS, or HL)
2. Conclude that any corresponding parts are equal — for example, $\overline{AC} = \overline{DF}$ or $\angle C = \angle F$

CPCTC is the payoff of a congruence proof. You do the work of establishing congruence so that you can then claim specific sides or angles are equal.

Quick Reference: Which Criterion to Use

Worked Examples

Example 1: Identifying SSS

Two triangles have the following side lengths:

• $\triangle ABC$: $AB = 7$, $BC = 9$, $AC = 12$
• $\triangle DEF$: $DE = 7$, $EF = 9$, $DF = 12$

All three pairs of corresponding sides are equal ($7 = 7$, $9 = 9$, $12 = 12$).

Answer: $\triangle ABC \cong \triangle DEF$ by SSS.

Example 2: Identifying SAS

In $\triangle PQR$ and $\triangle XYZ$:

• $PQ = XY = 5$
• $\angle Q = \angle Y = 60\degree$
• $QR = YZ = 8$

The angle $\angle Q$ is between sides $PQ$ and $QR$, and $\angle Y$ is between sides $XY$ and $YZ$. Two sides and the included angle match.

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

Example 3: Using CPCTC to find a missing side

Given that $\triangle ABC \cong \triangle DEF$ by ASA, where $\angle A = \angle D = 50\degree$, $AB = DE = 10$, and $\angle B = \angle E = 70\degree$. Find $DF$.

Step 1: Since the triangles are congruent by ASA, all corresponding parts are equal (CPCTC).

Step 2: Side $AC$ corresponds to side $DF$.

Step 3: Find $\angle C$:

$\angle C = 180\degree - 50\degree - 70\degree = 60\degree$

Step 4: Using the law of sines to find $AC$:

$\frac{AC}{\sin B} = \frac{AB}{\sin C}$

$\frac{AC}{\sin 70\degree} = \frac{10}{\sin 60\degree}$

$AC = \frac{10 \times \sin 70\degree}{\sin 60\degree} = \frac{10 \times 0.9397}{0.8660} \approx 10.85$

Step 5: By CPCTC, $DF = AC \approx 10.85$.

Answer: $DF \approx 10.85$.

Example 4: HL in a right triangle

Two right triangles have:

• Hypotenuse of each = 13
• One leg of each = 5

By the Pythagorean theorem, the other leg of each triangle is:

$\text{other leg} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$

Since the hypotenuse and one leg are equal, the triangles are congruent by HL. By CPCTC, the other leg of both triangles is $12$.

Example 5: Why SSA fails here

Consider two triangles where $AB = DE = 8$, $BC = EF = 5$, and $\angle A = \angle D = 30\degree$. Is $\triangle ABC \cong \triangle DEF$?

The known angle ($\angle A$) is not between the two known sides ($AB$ and $BC$). This is the SSA arrangement.

With $\angle A = 30\degree$, side $AB = 8$, and side $BC = 5$: the side opposite the $30\degree$ angle ($BC = 5$) is less than the other given side ($AB = 8$). In this configuration, there are two possible triangles — one where $\angle C$ is acute and another where $\angle C$ is obtuse.

Answer: We cannot conclude congruence. SSA is ambiguous, and the given information does not determine a unique triangle.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Congruent triangles have the same shape and size — all corresponding sides and angles are equal
• Five valid criteria prove congruence: SSS, SAS, ASA, AAS, and HL (right triangles only)
• SSA is NOT valid — two sides and a non-included angle can produce two different triangles
• CPCTC is the conclusion, not the method: first prove congruence, then use CPCTC to claim corresponding parts are equal
• The order of vertices in $\triangle ABC \cong \triangle DEF$ tells you which parts correspond: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$

Return to Geometry for more topics in this section.