Triangle angle sum is one of 5 triangles formulas in the geometry section of this library, and it is used at middle school level.
Why triangle angle sum works
Draw a line through one corner parallel to the opposite side. The two angles that spill out on either side of that corner match the triangle's other two angles, because parallel lines create equal alternate angles, and the three of them together lie along a straight line.
What each symbol means
$A,B,C$ are the interior angles.
Triangle angle sum: when it holds
This is for a triangle in a flat Euclidean plane.
When it stops applying
A flat surface is the condition. On the curved surface of a globe you can draw a triangle with a corner at the north pole and two corners on the equator whose three angles are all right angles, giving a total of 270 degrees.
Triangle angle sum: a worked example
If $A=50^\circ$ and $B=60^\circ$, then $C=70^\circ$.
The mistake to avoid
What people do: Subtracting only one known angle from 180 when two are given.
Why it goes wrong: Both known angles have to be taken off, so with 50 and 60 degrees given, the missing angle is 70 degrees rather than 130.
Do this instead: Add the two known angles together first, then subtract that total from 180.
Triangle angle sum: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A,B,C$ are the interior angles.
- Check the conditions before substituting. This is for a triangle in a flat Euclidean plane.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Triangles
- Level
- Middle school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where triangle angle sum comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Pythagorean Theorem — the lesson behind this formula: relate the side lengths of a right triangle.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about triangle angle sum
Can a flat triangle have two right angles?
No. Two right angles already use up the whole 180 degrees, leaving nothing at all for the third corner.
What is the total in radians?
It is π radians. Half a turn is the same amount of rotation whether you name it 180 degrees or π.
How does this give the exterior angle result?
An exterior angle and the interior angle beside it make a straight line, so the exterior angle must equal the two far interior angles added together.
Must an equilateral triangle have 60-degree angles?
Yes. Three equal angles sharing a total of 180 degrees leaves each of them at exactly 60.