The sum of angles in a triangle · 마틴과 코딩하기

Why the angles of a triangle always make 180° and how to work out the third angle.

Three angles, always 180°

Draw any triangle — thin, wide, wonky. Measure its three angles and add them. You get 180°. Try another triangle. 180° again. And again. Always 180°.

This is one of the most famous rules in all of geometry: the sum of the interior angles of a triangle is 180° — exactly a straight angle, a straight line.

You can prove it with your hands: draw a triangle on paper, cut it out and tear off all three corners. Put the torn corners vertex to vertex, next to each other. They make a straight line. The three angles of any triangle together pave a straight line.

What is it good for? For the same magic as last time: you do not have to measure everything. You know two angles → you work out the third: 180 minus those two. The triangle tells you the third angle itself.

And it is also a detector rule: a triangle with angles 90°, 60° and 45° simply does not exist. The sum 195° gives it away.

In class they are drawing triangles. Jonáš measures his: 74°, 62° and at the third angle he struggles — the paper is crumpled at the vertex and the protractor slips. Ema watches him: “So do not measure it. Calculate it.” Jonáš raises an eyebrow. “The sum is always 180. You have 74 and 62, that is 136. To one hundred and eighty there is left…” Jonáš works it out: “44!” To be sure he smooths the corner and measures again: the protractor shows 44°, maybe a hair more. “The calculation is more precise than my measuring,” he admits. From then on nobody in the row measures the third angle.

💡

Do the tearing proof for real — a triangle cut out and put together once you remember for life better than ten rules.

Working out the third angle

The method is two steps:

Add the two known angles. For example 74° + 62° = 136°.
Subtract from 180°. 180 − 136 = 44°. The third angle is 44°.

Written as a formula: γ = 180° − α − β.

A sense check: the result must be bigger than 0° (otherwise the triangle would not close) and smaller than 180°. And a fast visual check: did an acute angle come out at a pointy corner? Obtuse at a slumped one?

Special triangles this rule runs:

Right-angled (one angle 90°): for the other two exactly 90° is left. That is why both are acute.
Equilateral (all sides and angles the same): 180 : 3 = 60° each. Always, no exception.
Isosceles (two same sides): the angles at the base are the same. You know the vertex angle 40°? For two equal ones 140° is left, so 70° each.

A detector of impossible triangles: a sum other than 180° = such a triangle does not exist. Two right angles in one triangle? 90 + 90 = 180 and nothing is left for the third — it cannot. Two obtuse? Even worse. In a triangle there is always at most one right or obtuse angle.

Work out the third angle: 74° and 62°

Step 1: I add the known ones: 74 + 62 = 136°.

Step 2: I subtract from 180: 180 − 136 = 44°.

Check: 74 + 62 + 44 = 180 ✓. And sense: all three angles are acute, such a triangle happily exists (it is called acute-angled).

That is exactly what Jonáš counted in the story — and the calculation beat crumpled paper.

A right-angled triangle with an angle of 35°

The triangle has a right angle (90°) and one angle 35°. What is the third?

180 − 90 − 35 = 55°.

A faster trick for right-angled triangles: the two acute angles split the remaining 90°. So 90 − 35 = 55. The same result, one step less.

Check: 90 + 35 + 55 = 180 ✓. And both remaining angles came out acute — in a right-angled triangle that must always be so.

Does a triangle with angles 100°, 45° and 45° exist?

I add: 100 + 45 + 45 = 190°.

That is more than 180° → such a triangle does not exist. If you tried to draw it, the arms would never meet so the angles sit — the triangle would not “close”.

A fix so it exists: knock the obtuse angle down to 90° (90 + 45 + 45 = 180 ✓ — an isosceles right-angled triangle, half a square). The sum 180° is a law: what breaks it, you will not find on paper.

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Do not forget that the rule holds for INTERIOR angles. When in a question you see an angle drawn outside the triangle (beyond an extended side), first convert it to an interior one — they are adjacent angles, so interior = 180° minus the outer one. Only then add to 180°.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.

Work out the third angle of a triangle: 180 minus the two given. Write the answer as a number (degrees with no mark).

Practise the questions

On paper

시작 with scissors, finish with calculations.

The tearing proof: draw a big triangle, cut it out, tear off the corners and put them vertex to vertex. Stick the result in your notebook — it should make a straight line.
Work out the third angle: 60° and 70°; 90° and 28°; 115° and 40°. By each one write a check sum.
An isosceles triangle has 36° at the vertex. What are the angles at the base? And what angles does an equilateral triangle have?
Decide which triangles exist: (80°, 60°, 40°), (90°, 90°, 0°), (55°, 55°, 70°). By the ones that do not exist write why.

Now you 💪

I can work out the third angle from two given ones and check with the sum 180°.
I know why a triangle cannot have two right angles.
I know by heart the angles of an equilateral triangle and the rest to 90° in a right-angled one.

✅

Done when: You use the 180° rule: you work out the third angle, you spot an impossible triangle, and you have seen the proof with your own hands.

What to take from this lesson

The sum of the interior angles of every triangle is exactly 180°.
The third angle = 180° minus the two known. A calculation is often more precise than measuring.
Equilateral triangle: 3 × 60°. Right-angled: the two acute angles add to 90°.
A sum other than 180° = the triangle does not exist. At most one obtuse or right angle.
The tearing proof: three torn corners put together make a straight line.