Key Points

Figures that have exactly the same shape and size are called congruent. They can be perfectly superimposed on each other, which means one fits exactly over the other.

Two geometric figures are congruent if they have the exact same shape and size. One can be perfectly superimposed on the other, which may require rotation or flipping.

Two triangles are congruent if the three sides of one triangle are equal to the three corresponding sides of the other triangle. For example, if in $\triangle ABC$ and $\triangle PQR$, $AB = PQ$, $BC = QR$, and $AC = PR$, then $\triangle ABC \cong \triangle PQR$.

The Side-Side-Side (SSS) condition states that two triangles are congruent if the three sides of one triangle are equal in length to the three corresponding sides of the other triangle.

The Side-Angle-Side (SAS) condition states that two triangles are congruent if two sides and the angle included between them in one triangle are equal to the corresponding two sides and included angle in the other triangle.

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

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

The Angle-Side-Angle (ASA) condition states that two triangles are congruent if two angles and the side included between them in one triangle are equal to the corresponding two angles and included side in the other triangle.

Two triangles are congruent if any two pairs of angles and one pair of corresponding non-included sides are equal. This is valid because if two angles are known, the third angle is also fixed as the sum is $180^\circ$.

The Angle-Angle-Side (AAS) condition states that two triangles are congruent if two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle.

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. This rule applies only to right-angled triangles.

The Right Angle-Hypotenuse-Side (RHS) condition applies to right-angled triangles. Two right 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.

The AAA (Angle-Angle-Angle) and SSA (Side-Side-Angle) conditions are not sufficient to prove that two triangles are congruent. AAA only proves similarity (same shape), and SSA can lead to ambiguous cases.

The AAA (Angle-Angle-Angle) and SSA (Side-Side-Angle) conditions are not sufficient to prove congruence. AAA only proves similarity (same shape, different size), and SSA can result in two different possible triangles.

The order of vertices in a congruence statement is important. $\triangle \mathrm{ABC} \cong \triangle \mathrm{XYZ}$ means A corresponds to X, B corresponds to Y, and C corresponds to Z, which implies corresponding parts are equal.

The order of vertices in a congruence statement is crucial as it shows the correspondence. Writing $\triangle ABC \cong \triangle PQR$ implies that vertex $A$ corresponds to $P$, side $AB$ corresponds to side $PQ$, and $\angle A$ corresponds to $\angle P$.

If two triangles are proven to be congruent, then their corresponding parts (sides and angles) are also equal. This property is often abbreviated as CPCTC.

If two triangles are proven to be congruent, then their corresponding parts (sides and angles) are equal. For example, if $\triangle \mathrm{ABC} \cong \triangle \mathrm{DEF}$, then $\mathrm{AB} = \mathrm{DE}$, $\mathrm{BC} = \mathrm{EF}$, $\mathrm{AC} = \mathrm{DF}$, $\angle \mathrm{A} = \angle \mathrm{D}$, $\angle \mathrm{B} = \angle \mathrm{E}$, and $\angle \mathrm{C} = \angle \mathrm{F}$.

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

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

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

An equilateral triangle has all three sides of equal length. Consequently, all three of its interior angles are also equal, and each angle measures exactly $60^\circ$.

The concept of congruence applies to all geometric figures. For example, two circles are congruent if their radii are equal, and two squares are congruent if their side lengths are equal.