The angles of a triangle follow three rules: the three interior angles always add to 180°, each exterior angle equals the sum of the two opposite interior angles, and the largest angle always sits opposite the longest side. To find a missing angle, subtract the two angles you know from 180° — for example, angles of 90° and 36.87° leave 180 − 90 − 36.87 = 53.13°.
Triangle angles are one of the most reliable topics in school mathematics, because the rules never change and the check is instant: if your three angles do not add to 180°, something is wrong. Yet students still lose marks — usually by forgetting the exterior-angle rule, or by assuming the angles split evenly when they do not.
Rule 1: The Three Interior Angles Add to 180°
A + B + C = 180°. Always, for every flat triangle, no exceptions. This is the master rule, and it is the one that finds missing angles: if you know two angles, the third is 180° minus their sum.
Worked example: the 3-4-5 triangle. The angles are 90°, 36.87° and 53.13°: 90 + 36.87 + 53.13 = 180°. The right angle is opposite the longest side (5), 36.87° opposite side 3, 53.13° opposite side 4 — smallest side, smallest angle.
Worked example: find the missing angle. A triangle has angles of 50° and 60°. The third is 180 − 50 − 60 = 70°. Check: 50 + 60 + 70 = 180°. Done in one line — and that one line is the most common exam question in the whole topic.
Worked example: two equal angles. A triangle has one angle of 40° and the other two equal. The equal pair shares 180 − 40 = 140°, so each is 70°. This is an isosceles triangle with vertex angle 40° and base angles 70° each.
Worked example: angles in a ratio. A triangle’s angles are in the ratio 1 : 2 : 3. The parts add to 6, so each part is 180/6 = 30°, and the angles are 30°, 60° and 90° — the 30-60-90 triangle.
Check any triangle instantly with a free triangle calculator, which returns all three angles and confirms they add to 180°.
Rule 2: The Exterior Angle Theorem
Each exterior angle equals the sum of the two opposite interior angles. Extend one side of a triangle and the angle outside, next to the vertex, is the exterior angle. It is always equal to the two interior angles that are not next to it — the “remote interiors”. It is also supplementary to the interior angle at the same vertex: exterior + interior = 180°, because they sit on a straight line.
Worked example: angles 50°, 60°, 70°. Extend the side next to the 70° angle. The exterior angle is 50 + 60 = 110°. Check the other way: 180 − 70 = 110°. Both routes agree — and when they do not, one of your interior angles is wrong.
Worked example: the 3-4-5 triangle. The exterior angle at the 90° vertex is 36.87 + 53.13 = 90°. At the 36.87° vertex the exterior is 90 + 53.13 = 143.13°, and at the 53.13° vertex it is 90 + 36.87 = 126.87°. Add all three: 90 + 143.13 + 126.87 = 360° — the exterior angles of any triangle always add to 360°.
Why this rule matters: it turns a two-step problem into a one-step problem — if a question gives you the two remote interiors, you just add. It also runs backwards: an exterior angle of 110° with one remote interior of 50° leaves 110 − 50 = 60° for the other.
Rule 3: Largest Angle Opposite Longest Side (and Smallest Opposite Shortest)
The longest side is always opposite the largest angle; the shortest side opposite the smallest angle. This rule is the consistency check for every triangle answer. If your working ever gives the largest angle opposite the shortest side, the working is wrong — full stop.
Worked example: sides 7, 8, 9. The angles are 48.19° (opposite 7), 58.41° (opposite 8) and 73.40° (opposite 9) — same order: 7 < 8 < 9 and 48.19 < 58.41 < 73.40. The triangle is acute, all angles under 90°.
Worked example: sides 2, 3, 4. The angles are 28.96° (opposite 2), 46.57° (opposite 3) and 104.48° (opposite 4) — the largest angle is obtuse, because side 4 is long relative to the other two.
Worked example: sides 5, 5, 6. The angles are 53.13°, 53.13° and 73.74°. Two equal sides give two equal angles — the isosceles rule is just Rule 3 applied twice: equal sides, equal opposite angles.
Shortcuts for Special Triangles
Equilateral: all angles 60°. Three equal sides force three equal angles, and three equal angles sharing 180° must each be 60°. No calculation needed — if a triangle is equilateral, you know every angle before you start.
Isosceles: the base angles are equal. Two equal sides mean two equal angles, opposite those sides. Find the vertex angle first, then split the remainder: base angle = (180° − vertex angle)/2. Vertex 40° gives base angles of (180 − 40)/2 = 70° each; vertex 100° gives 40° each. And backwards: base angles of 70° give a vertex angle of 180 − 70 − 70 = 40°.
Right triangle: the two acute angles add to 90°. In a 3-4-5 triangle they are 36.87° and 53.13°. If you know one acute angle of a right triangle, the other is 90° minus it.
30-60-90: the angles are 30°, 60° and 90°. The sides are in the ratio 1 : √3 : 2 — with hypotenuse 10, the legs are 5 and 8.6603. 45-45-90: the angles are 45°, 45° and 90°. The legs are equal — with hypotenuse 8, each leg is 8/√2 = 5.6569, and the area is ½ × 5.6569 × 5.6569 = 16.
How to Find a Missing Angle, Step by Step
Problem: a triangle has angles of 50° and 60°. Find the third angle.
- Write the rule: A + B + C = 180°.
- Substitute what you know: 50 + 60 + C = 180.
- Add the known angles: 50 + 60 = 110.
- Subtract from 180: C = 180 − 110 = 70°.
- Check: 50 + 60 + 70 = 180° ✓.
Problem: an isosceles triangle has a vertex angle of 100°. Find the base angles.
- The base angles are equal — call each one x.
- Write the rule: 100 + x + x = 180.
- Subtract: 2x = 180 − 100 = 80.
- Divide: x = 40°. The angles are 100°, 40°, 40° — check: 100 + 40 + 40 = 180° ✓.
Problem: one angle is twice another, and the third is 90°. Find all three.
Call the angles x, 2x and 90°: x + 2x + 90 = 180, so 3x = 90 and x = 30°. The angles are 30°, 60° and 90° — the 30-60-90 triangle again. Check: 30 + 60 + 90 = 180° ✓.
4 Mistakes That Cost Marks
1. Assuming the angles split evenly. Three angles that add to 180° are not 60° each unless the triangle is equilateral. A triangle with angles 50°, 60° and 70° adds to 180° without any two being equal.
2. Using the wrong pair for the exterior angle. The exterior angle equals the two opposite interiors — not the adjacent one. At the 70° vertex, the exterior is 50 + 60 = 110°, not 70 + anything. The adjacent interior angle is the one the exterior is supplementary to (70 + 110 = 180), never one of the two you add.
3. Forgetting that the angles must add to 180°. If your three angles add to 179° or 181°, one of them is wrong — usually a rounding error. Compute the third angle from 180° minus the other two, rather than estimating it separately.
4. Mixing up which angle is opposite which side. The angle opposite side a is A, opposite b is B, opposite c is C. In a 3-4-5 triangle the 90° angle is opposite the 5 — if your working puts 90° opposite the 3, the labelling is wrong and every angle that follows will be wrong too. The triangle angle rules work only with consistent labelling: longest side, largest angle.
Why 180°? (The One-Paragraph Proof)
Draw a line through one vertex parallel to the opposite side. The three angles at that vertex sit on a straight line, so they add to 180° — and the two outer ones are alternate interior angles, equal to the triangle’s other two angles. So the triangle’s three angles add to 180°. The rule fails on a sphere, where parallel lines do not exist and a triangle’s angles can add to more than 180°.
Frequently Asked Questions
Do the angles of a triangle add up to 180 degrees?
Yes — the three interior angles of any flat triangle add to exactly 180°. For example, the 3-4-5 triangle has angles 90°, 36.87° and 53.13°, and 90 + 36.87 + 53.13 = 180°. If your angles do not add to 180°, at least one is wrong.
What is the exterior angle of a triangle?
The angle formed by extending one side of the triangle. It equals the sum of the two opposite interior angles: for a triangle with angles 50°, 60° and 70°, the exterior angle at the 70° vertex is 50 + 60 = 110°. It is also supplementary to the adjacent interior angle (70 + 110 = 180).
How do you find a missing angle in a triangle?
Subtract the two known angles from 180°. A triangle with angles of 50° and 60° has a third angle of 180 − 50 − 60 = 70°. For an isosceles triangle, split the remainder equally: a vertex angle of 40° leaves (180 − 40)/2 = 70° for each base angle.
What are the angles of a 3-4-5 triangle?
90°, 36.87° and 53.13°. The right angle is opposite the longest side (5). The other two are arcsin(3/5) = 36.87° and arcsin(4/5) = 53.13°. The two acute angles add to 90°, as they must in any right triangle.
What are the angles of an equilateral triangle?
All 60° — three equal sides force three equal angles sharing 180°. An isosceles triangle has two equal angles instead — the base angles.
Can a triangle have two right angles?
No. Two right angles already add to 180°, leaving 0° for the third angle — which is not a triangle at all, but a straight line. For the same reason a triangle can have at most one obtuse angle.
Keep Practicing
Work these by hand, then verify each with the free triangle calculator: angles 50° and 60° → third angle 70°; vertex angle 100° isosceles → base angles 40° each; angles in ratio 1 : 2 : 3 → 30°, 60°, 90°; exterior angle at the 70° vertex → 110°; 3-4-5 → 90°, 36.87°, 53.13°. If your angles add to 180° and the largest sits opposite the longest side, you can trust the answer.