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In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The theorem is written as $c^2 = a^2 + b^2$ where $c$ is the hypotenuse.

Two triangles are similar if their corresponding angles are congruent and the lengths of their corresponding sides are proportional. This can be identified through Angle-Angle (AA), Side-Side-Side (SSS), or Side-Angle-Side (SAS) criteria.

The Law of Sines states that in any triangle, the ratios of the length of a side to the sine of its opposite angle are equal. This can be written as $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, where $a$, $b$, and $c$ are sides of the triangle and $A$, $B$, and $C$ are the opposite angles.

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This can be written as $a + b > c$, $a + c > b$, and $b + c > a$ for a triangle with sides $a$, $b$, and $c$.

Two triangles are congruent if they satisfy any one of the following conditions: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), or Angle-Angle-Side (AAS).

For any triangle, the measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. This can be written as $\text{ext} \angle = \angle A + \angle B$ for an exterior angle at vertex C with interior angles $A$ and $B$.

In an isosceles triangle, the base angles—that is, the angles opposite the two equal sides—are equal to each other.