Equilateral Triangle: All three sides are of equal length, and all three angles are equal (60 degrees each in the case of an equilateral triangle).

Isosceles Triangle: Two sides are of equal length, and the corresponding angles opposite those sides are equal.

Scalene Triangle: All three sides and angles are of different lengths and measures.

Acute Triangle: All three angles are less than 90 degrees.

Obtuse Triangle: One angle is greater than 90 degrees.

Right Triangle: One angle is exactly 90 degrees.

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean Theorem is expressed as \(c^2 = a^2 + b^2\), where c is the length of the hypotenuse, and aa and bb are the lengths of the other two sides.

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

The centroid is the point where the medians of the triangle intersect.

The circumcenter is the point where the perpendicular bisectors of the sides intersect.

The incenter is the point where the angle bisectors of the triangle intersect.

The orthocenter is the point where the altitudes of the triangle intersect.

The area of a triangle can be calculated using various formulas, such as \(\frac{1}{2} \times \text{base} \times \text{height}\) or by using Heron's formula, which involves the lengths of all three sides.

Two triangles are similar if their corresponding angles are equal, and their corresponding sides are in proportion.

Two triangles are congruent if their corresponding angles and sides are equal.