Triangle Types and Rules

Triangles are those handy‐dandy, three‐sided polygons with which mathematicians have done so much work, particularly the right triangle.

The most important thing to remember about all triangles is that the sum of their interior angles is 180°.

• Again, the sum of the interior angles of any triangle is 180°.
• Once more, the sum of the interior angles of any triangle is 180°!

In all three of the following triangles look different, but the sum of their angles a° + b° + c° = 180°

Angles and their Side Lengths

The opposite side length from the angle is defined by the associated angle.

In the following diagram, ∠a defines side A, ∠b defines side B, and ∠c defines side C. This means that if the measure of ∠a increases, the length of side A will increase as well; if the measure of ∠a decreases, the length of side A will decrease. The same is true for ∠b and ∠c.

Equilateral Triangle

An equilateral triangle is a triangle in which the value of each interior angle is the same and the length of each side is the same, too.

All interior angles of the equilateral triangle have the degree value 60° and all the side lengths have length y.

Isosceles Triangle

An isosceles triangle is a triangle in which two of the three angles are the same (called base angles) and two corresponding sides have the same length. The third angle, which is not equal to the base angles, is called the vertex angle. In this case, ∠c is the vertex angle, while ∠a and ∠b are equal to each other and sides A and B are equal to each other.

Scalene Triangle

A scalene triangle is a triangle in which all three angles and side lengths are unique. 

Thus, none of the angles or sides are equivalent: ∠x ≠ ∠y ≠ ∠z and X ≠ Y ≠ Z.          

Right Triangle 

A right triangle is a triangle in which two sides are perpendicular, forming a 90° angle (a right angle).

The third side is known as the hypotenuse.

In the above triangle, ∠a and ∠b are complementary angles, and C is the hypotenuse of the right triangle.

Practice Problems

1. What is the sum of the degree value of two triangles?
2. If a triangle is equilateral, what do you know about the three side lengths?
3. If a triangle is a right triangle, which of the following must be TRUE?

• A) All three interior angles sum to 90°
• B) Two of the interior angles are complementary
• C) ll three interior angles are equivalent
• D) One interior angle is equal to 90°

Answers to Practice Problems

1. 360°
2. All three side lengths are equal
3. D