Triangle angles and classification
Triangles are the simplest polygons and the building blocks of geometry and engineering. Their angles always follow strict rules.
Classifying by sides
A scalene triangle has no congruent sides, an isosceles triangle has at least two congruent sides, and an equilateral triangle has three congruent sides. Every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.
Classifying by angles
An acute triangle has three acute angles, a right triangle has one right angle, and an obtuse triangle has one obtuse angle. An equiangular triangle has three congruent angles, each 60 degrees, and is also equilateral.
The triangle angle sum theorem
The interior angles of any triangle add to 180 degrees. If two angles measure 48 and 67 degrees, the third measures 180 - 115 = 65 degrees. A triangle can have at most one right or obtuse angle, and the two acute angles of a right triangle are complementary.
Exterior angles
An exterior angle is formed by one side of a triangle and the extension of an adjacent side. The exterior angle theorem says it equals the sum of the two remote interior angles. If the remote interior angles are 40 and 75 degrees, the exterior angle is 115 degrees, which is also 180 minus the adjacent interior angle of 65.
Angles from algebra
If a triangle's angles are x, 2x, and 3x, then 6x = 180, so x = 30 and the angles are 30, 60, and 90 degrees, a right triangle. Setting up the sum to 180 is the standard first step.
Triangles in structures
Triangles are rigid: once the side lengths are fixed, the shape cannot change, unlike a rectangle frame, which can collapse into a parallelogram. That is why bridges, roof trusses, cranes, and bicycle frames use triangular bracing.
Interior and exterior angle facts
At each vertex, an interior angle and its exterior angle form a linear pair, so they add to 180 degrees. Taking one exterior angle at each vertex, the three exterior angles of any triangle add to 360 degrees.
Worked example
In right triangle ABC, angle C is 90 degrees and angle A is 3 times angle B. Since A and B are complementary, 3B + B = 90, so B = 22.5 degrees and A = 67.5 degrees. The exterior angle at C measures 90 degrees, equal to A + B as the exterior angle theorem predicts.
Key takeaways
Triangles are classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse, equiangular). Interior angles always sum to 180 degrees, so a triangle has at most one right or obtuse angle, and a right triangle's acute angles are complementary. An exterior angle equals the sum of the two remote interior angles, and the exterior angles, one per vertex, sum to 360 degrees. Triangles are rigid, which is why they brace structures.
Flashcard
No congruent sides
Check your understanding
A triangle has angles of 48 and 67 degrees. What is the third angle?