Triangles: Types and Properties

Triangle Properties

By the end of this lesson, you will be able to classify triangles, apply the angle sum property, and use the triangle inequality.

Classifying Triangles

By angles:

• Acute: all angles less than 90°
• Right: one angle equals exactly 90°
• Obtuse: one angle greater than 90°

By sides:

• Equilateral: all three sides equal; all angles = 60°
• Isosceles: at least two sides equal; base angles are equal
• Scalene: no sides equal; no angles equal

The Angle Sum Property

The three interior angles of any triangle sum to 180°.

$$\angle A + \angle B + \angle C = 180°$$

Example: A triangle has angles 45° and 75°. Find the third angle.

$\angle C = 180° - 45° - 75° = 60°$

Example: An isosceles triangle has a base angle of 50°. Find all angles.

The two base angles are equal: $50° + 50° + \angle C = 180°$ → $\angle C = 80°$.

Exterior Angles

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles:

$\angle \text{exterior} = \angle A + \angle B$ (the two angles not adjacent to the exterior angle)

Example: A triangle has angles 40°, 60°, 80°. An exterior angle at the 80° vertex is $180° - 80° = 100°$.

Check: $40° + 60° = 100°$ ✓

Triangle Inequality

The sum of any two sides must be greater than the third side.

$$a + b > c \quad a + c > b \quad b + c > a$$

Example: Can 3, 5, 9 form a triangle?

$3 + 5 = 8 < 9$ ← FAILS. These cannot form a triangle.

Example: Can 4, 6, 8 form a triangle?

$4 + 6 = 10 > 8$ ✓; $4 + 8 = 12 > 6$ ✓; $6 + 8 = 14 > 4$ ✓ → Yes.

Special Triangles

30-60-90 triangle: sides in ratio $1 : \sqrt{3} : 2$

45-45-90 triangle: sides in ratio $1 : 1 : \sqrt{2}$ (isosceles right triangle)

These will be used extensively in trigonometry.

Summary

Angle sum: $\angle A + \angle B + \angle C = 180°$.

Exterior angle: = sum of the two non-adjacent interior angles.