Key Points

Two triangles are congruent if they have the same shape and size, meaning their corresponding sides and angles are equal. The symbol for congruence is $\cong$.

Two triangles are congruent if their corresponding sides and corresponding angles are equal. If $\triangle \mathrm{ABC}$ is congruent to $\triangle \mathrm{PQR}$, we write it as $\triangle \mathrm{ABC} \cong \triangle \mathrm{PQR}$.

If two triangles are proven to be congruent, then their corresponding parts (sides and angles) are equal. This is abbreviated as CPCT and used to deduce equalities after proving congruence.

The order of vertices in a congruence statement is crucial. $\triangle \mathrm{ABC} \cong \triangle \mathrm{PQR}$ implies the correspondence $\mathrm{A} \leftrightarrow \mathrm{P}$, $\mathrm{B} \leftrightarrow \mathrm{Q}$, and $\mathrm{C} \leftrightarrow \mathrm{R}$, meaning $\mathrm{AB} = \mathrm{PQ}$, $\angle \mathrm{B} = \angle \mathrm{Q}$, etc.

Two triangles are congruent if two sides and the included angle of one triangle are equal to the two corresponding sides and the included angle of the other triangle. The angle must be between the two sides.

CPCT stands for 'Corresponding Parts of Congruent Triangles are equal'. After proving two triangles are congruent, this reason is used to state that their remaining corresponding parts are also equal.

Two triangles are congruent if two angles and the included side of one triangle are equal to the two corresponding angles and the included side of the other triangle. The side must be between the two angles.

Side-Angle-Side (SAS) rule: Two triangles are congruent if two sides and the included angle of one triangle are equal to the two corresponding sides and the included angle of the other triangle.

Angle-Side-Angle (ASA) rule: Two triangles are congruent if two angles and the included side of one triangle are equal to the two corresponding angles and the included side of the other triangle.

Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. The side does not need to be between the angles.

Two triangles are congruent if the three sides of one triangle are equal to the three corresponding sides of the other triangle.

Angle-Angle-Side (AAS) rule: Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. The side does not need to be included between the angles.

Side-Side-Side (SSS) rule: Two triangles are congruent if the three sides of one triangle are equal to the three corresponding sides of the other triangle.

Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of the other triangle.

The SSA (Side-Side-Angle) and ASS (Angle-Side-Side) conditions are not sufficient to prove triangle congruence. Also, AAA (Angle-Angle-Angle) proves similarity, not congruence, as triangles can have the same angles but different side lengths.

Right angle-Hypotenuse-Side (RHS) rule: Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.

In an isosceles triangle, angles opposite to the equal sides are equal. If in $\triangle ABC$, side $AB = AC$, then $\angle C = \angle B$.

Equality of all three angles (AAA) is not sufficient for congruence, as it only proves similarity. Also, Side-Side-Angle (SSA or ASS) is not a valid congruence rule.

In a triangle, sides opposite to equal angles are equal. If in $\triangle ABC$, $\angle B = \angle C$, then side $AC = AB$.

In an isosceles triangle, the angles opposite to the equal sides are equal. If in $\triangle \mathrm{ABC}$, $\mathrm{AB} = \mathrm{AC}$, then $\angle \mathrm{C} = \angle \mathrm{B}$.

In a triangle, the sides opposite to equal angles are equal. If in $\triangle \mathrm{ABC}$, $\angle \mathrm{B} = \angle \mathrm{C}$, then the sides opposite to them, $\mathrm{AC}$ and $\mathrm{AB}$, are equal.

An equilateral triangle has all three sides equal, and consequently, all three interior angles are equal to $60^\circ$.

An equilateral triangle has all three sides equal. As a result, all three of its interior angles are also equal, and each measures $60^\circ$.

When writing a congruence relation, the order of vertices must match the corresponding equal parts. $\triangle ABC \cong \triangle PQR$ means vertex A corresponds to P, B to Q, and C to R.