Concept 5 of 6
Angle Sum: The 180° Triangle Rule
Today's big idea
Every triangle's three angles add to 180°, so any two of them give you the third.
Stretch or squash the triangle however you like, the three angles still total 180°.
How it works
👀 What's going on?
Draw any triangle you like. Big, thin, tilted, it makes no difference at all. Its three angles will always add to exactly 180°, and that one fact unlocks a surprising number of problems.
💡 Here's the trick
Look at the triangle above. Its angles are 50°, 60° and 70°, which come to 180°. Tear the three corners off a paper triangle and lay their tips together, and they line up along a straight edge, which is 180°. So knowing any two angles hands you the third.
🪄 How to do it
- 1.Add the two angles you already know, the way the picture above adds 50° and 60°.
- 2.Subtract that sum from 180°.
- 3.Whatever is left over is the missing angle.
✍️ Try it
One triangle has angles of 55° and 70°. Another has a right angle and 25°. Find the third angle in each of them.
The first gives 180 − 125 = 55°, matching an angle it already had, so that triangle is isosceles. The second has a right angle worth 90°, so 180 − 115 = 65°.
Two triangles are worth memorising. An equilateral triangle has three 60° angles. A right-angled isosceles triangle has 90°, 45° and 45°.
⚠️ Watch out
You might be thinking a bigger triangle must have bigger angles. It does not. Scale a triangle up to the size of a field and its angles never change. The 180° gets shared out however long the sides grow.
🌟 Why this matters
Roof frames, bridge trusses and folding chairs are all built from triangles, and their angles are what keep them rigid. Two known angles are enough to work out where to cut the third piece.
Exam tip: When a figure shows a right angle, one of your two known angles is already 90°.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Two angles of a triangle are 40° and 75°. Find the third.”
- 1
Recall the rule
All three triangle angles add to 180°. 📐
- 2
Sum the known angles
40 + 75 = 115
- 3
Subtract from 180
180 − 115 = 65
- 4
Answer ✓
Third angle = 65°.
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Angles in any triangle add to 180°. Add the two you know, then subtract from 180°.
TRIANGLE · ROUND 1 OF 3
Step 1 — add the two known angles: 50° + 60° = ?
Quick check
✅ VERIFICATION
Two angles of a triangle are 40° and 40°. What is the third angle?
⚡ Finish the activity above first to answer.
Keep going!
Up next: The Mirror Grid
Fold in half, do both sides match?