Triangles

A scalene triangle is a triangle in which all three sides have different lengths. In other words, it has no congruent sides. Additionally, in a scalene triangle, the three angles have different measures. In other words, scalene triangles have no congruent angles.

An isosceles triangle is a triangle that has exactly two sides of the same length. In other words, it has two congruent sides. The congruent sides are called legs while the third side is called the base. The angle between the legs is called the vertex angle and the other two angles are called base angles. In an isosceles triangle, the base angles are congruent angles — they have the same measure.

An equilateral triangle is a triangle in which the three sides have the same length. In other words, the three sides are congruent. In an equilateral triangle, the three angles have the same measure. Therefore, equilateral triangles are also equiangular triangles.

An acute triangle is a triangle where all angles are acute. In other words, the three angle measures are less than $90^{\circ }.$ In case the angles are measured in radians, a triangle is acute when each of the angle measures is less than $\frac{\pi }{2},$ or about $1.57$ radians.

An obtuse triangle is a triangle that has one obtuse angle. In other words, one of the angles measures more than $90^{\circ }.$ In case the angles are measured in radians, a triangle is obtuse when one of the angles measures more than $\frac{\pi }{2},$ or about $1.57$ radians.

A right triangle is a triangle that has one right angle. The side opposite the right angle is always the longest and is known as the hypotenuse. The other sides are commonly called legs. Notice that in a right triangle, the legs are perpendicular to each other. If one of the acute angles in the triangle is labeled, the legs can be discussed relative to that angle. Consider an acute angle labeled as $\angle \theta$ on the diagram. The side that forms the angle is the adjacent side and the side not touching the angle is the opposite side. Note that the opposite and adjacent sides change when $\angle \theta$ changes, but the hypotenuse is always the same.

An equiangular triangle is a triangle in which the three angles have the same measure. In other words, the three angles are congruent. In fact, the three angles measure $60^{\circ }$ each. If the angles are measured in radians, then the three measures are equal to $\frac{\pi }{3}.$ Notice that an equiangular triangle is also an acute triangle. In an equiangular triangle, the three sides have the same measure. Therefore, equiangular triangles are also equilateral triangles.