Triangle Classification & Angle Sum
Classify triangles by sides and by angles, find missing angles with the 180° sum, and shortcut with the exterior angle theorem.
1 Two ways to classify
Every triangle gets two labels — one for its sides, one for its angles.
By sides
Equilateral: all 3 sides equal. Isosceles: exactly/at least 2 equal (tick marks show it). Scalene: no equal sides.
By angles
Acute: all angles under 90°. Right: one 90° angle. Obtuse: one angle over 90°.
Classifications overlap. A triangle can be isosceles and acute, or scalene and right.
“Isosceles” describes the sides; “acute” describes the angles — they are not competing labels.
Quick check 1 — A triangle has sides 5, 5, 8. By sides it is…
Quick check 2 — A triangle has angles 50°, 60°, 70°. By angles it is…
2 Angle sum & exterior angle
The three interior angles always add to 180° — and the exterior angle is a shortcut around it.
a + b + c = 180°The exterior angle theorem
An exterior angle equals the sum of the two remote (non-adjacent) interior angles: 50° + 60° = 110°. No need to find the third angle first.
Angle sum is about angles, not sides. If you catch yourself adding side lengths to find
a missing angle, stop — the 180° belongs to the three interior angles.
Quick check 3 — Two angles of a triangle are 50° and 60°. The third angle is…
Quick check 4 — The remote interior angles are 50° and 60°. The exterior angle is…
Quick check 5 — Can 50°, 60°, 80° be the angles of a triangle?
3 Worked examples
From finding x to classifying with both labels.
Example 1 — solve for x, then the angles
- A triangle’s angles are (x + 10)°, 2x°, and (3x − 10)°.
- Angle sum: (x + 10) + 2x + (3x − 10) = 180.
- Combine: 6x = 180, so x = 30.
- The angles are 40°, 60°, 80° — check: 40 + 60 + 80 = 180. ✓
Example 2 — both classifications
- Sides 5, 5, 8: two equal → isosceles.
- Angles 50°, 65°, 65°: all under 90° → acute.
- Full description: isosceles and acute.
Example 3 — exterior angle backwards
- An exterior angle is 120°; one remote interior is 45°. Find the other.
- Exterior = sum of remotes: 45 + ? = 120.
- ? = 120 − 45 = 75°.