KS3 · Maths

Angle sum in a triangle

Draw any triangle, measure its three angles and add them up: 180°, every time. That isn't luck, and knowing why lets you find angles without a protractor.

Try to break the rule

Can you make a triangle whose angles don't add up to 180°?

angle A 63°. angle B 47°. angle C 70°. Classification: Seen for every triangle you try · not yet proved. Relationship: ∠A + ∠B + ∠C = 180°

angle A63°angle B47°angle C70°drag C, then B, and add the three angles

Seen for every triangle you try · not yet proved

∠A + ∠B + ∠C = 180°

Drag C along the faint line, and slide B along the base. Make the triangle tall and thin, short and wide, give it a right angle, or push C out past A so one angle goes over 90°. Every time, add up the three angles.

Watch out: Each angle is rounded to the nearest degree, so now and then your three readings total 179° or 181°. That's rounding, not a broken rule, and it's one more reason measuring can never make you certain.

Proof

Problem

Prove that the interior angles of any triangle ABC add up to 180°.

Your turn

Four triangles, one method

Find the missing angles. (a) Two angles are 126° and 45°. (b) An isosceles triangle has a top angle of 140°. (c) The angles are in the ratio 1 : 14 : 5. (d) The angles are x, x + 5 and x + 25 (in degrees).

(a) Write an equation from the angle sum: x + 126° + 45° = 180°.
missing step

Which line is step 2?

x = 180° − 171° = 9°x = 126° + 45° = 171°x = 180° − 126° = 54°

Name that triangle

What type of triangle is it? Is it even a triangle?

Each set of three angles claims to be a triangle. Add them up first, then decide where it belongs.

Still to sort

Equilateral (0)

All three angles are 60°.

Isosceles (0)

Exactly two of the angles are equal.

Scalene (0)

All three angles are different, and none is 90°.

Right-angled scalene (0)

One angle is 90°, and the other two are different.

Cannot be a triangle (0)

The three angles don't add up to 180°.

8 of 8 still to sort.

Watch out: Two equal angles can't rescue a set that doesn't add up to 180°. Check the total before you name the type.

What do you really think?

Jay's triangle

A triangle is drawn on the board. Two of its sides look the same length, but there are no marks on it and nothing is written about it. Jay measures its angles with a protractor, and they add up to 180°.

Which of these is closest to what you think?

It looks isosceles, so its two base angles must be equal.Jay has now proved that every triangle's angles add up to 180°.If Jay measured enough different triangles, that would prove it.Jay has shown it for this triangle, but can't call it isosceles.

How sure are you?

I'm sureI think soI'm guessing

It looks isosceles, so its two base angles must be equal.

This is a common misconception

Looks can fool you. You can only treat a triangle as isosceles if there are marks showing equal sides, you're told it's isosceles, or the angle facts show two equal angles. Here there's none of that, so you can't use 'the base angles are equal'.

Jay has now proved that every triangle's angles add up to 180°.

This is a common misconception

Jay has demonstrated it, for this one triangle. A proof has to work for every triangle, and that's what the parallel-lines proof does.

If Jay measured enough different triangles, that would prove it.

This is a common misconception

However many triangles you measure, there are always more you haven't tried, and protractor readings aren't exact anyway. More examples make the rule more convincing. Only a proof makes it certain.

Jay has shown it for this triangle, but can't call it isosceles.

This matches the evidence

Exactly right. The measurement demonstrates the angle sum for this one triangle (the proof covers the rest), and with no marks and no statement, you can't assume it's isosceles.

Work from what you're told or what you can prove, never from how a diagram looks or from a handful of examples.

Spot the slip

One wrong angle fact

Triangle ABC has a line OD through A, parallel to BC (O is on B's side, D is on C's side). Side BA is extended past A to F, and side CA is extended past A to G. You're told ∠GAF = 58° and ∠FAD = 71°. Find the three angles of triangle ABC, and say what type of triangle it is.

Sam's answer — which line goes wrong?

∠BAC = 58°, because it is vertically opposite ∠GAF.
∠ABC = 109°, because ∠ABC and ∠FAD are co-interior angles, so they add up to 180°.
∠ACB = 180° − 58° − 109° = 13°, because the angles in a triangle add up to 180°.
So the angles are 58°, 109° and 13°, which makes it an obtuse, scalene triangle.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Why every triangle's three angles make 180°, and how to use that to find the angles you're missing.

What you need to know

An interior angle is the angle inside a shape between two of its edges. A triangle has three vertices, so it has three interior angles.
The interior angles of any triangle add up to 180°, whether it is scalene, isosceles, equilateral, right-angled or obtuse.
Measuring triangles, or tearing off their corners, demonstrates the fact for the triangles you tried. A proof shows it for every triangle.
To find a missing angle, write an equation using the 180° total and solve it.
Don't decide a triangle's type from how it looks. You need marks on the diagram, a statement, or the angle facts.

The big picture

The three interior angles of any triangle add up to 180°. Measuring triangles demonstrates this, but only a proof shows it for every triangle, using angles on a straight line and alternate (or corresponding and vertically opposite) angles in parallel lines. To find a missing angle, write an equation from the 180° total and solve it: this works for two known angles, isosceles triangles, angles in a ratio and angles written with algebra. Knowing the angles tells you the triangle's type, and three angles that don't total 180° can't be a triangle.

Key points

1Angle sum: in every triangle ABC, ∠ABC + ∠BAC + ∠ACB = 180°.

2The proof: draw a line through one vertex parallel to the opposite side. The three angles on that line make 180°, and alternate angles show they equal the triangle's three angles.

3Parallel-line facts used: alternate angles are equal, corresponding angles are equal, co-interior angles add to 180°, vertically opposite angles are equal, angles on a straight line add to 180°.

4Isosceles: two equal base angles, so write 2 × (base angle) + (top angle) = 180°.

5Ratio: add up the parts, divide 180° by the total, then multiply back for each angle.

6Algebra: add the expressions, set the total equal to 180, solve, then work out each angle.

7Three angles can only make a triangle if they total 180°, so a triangle can never have two obtuse angles.

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Worked example

Problem

An isosceles triangle has its largest angle three times the size of each of its two equal angles. Find all three angles.

⚠ Watch out

Assuming a triangle is isosceles or right-angled because it looks like it. Unless there are marks showing equal sides or a right angle, or the question tells you, you can't assume it. Work only from what you're given.

🧠

Memory hook

Tear, line, 180: tear the three corners off any triangle and they fit side by side along a straight line, and a straight line is 180°.

✓

Check yourself

A triangle has angles of 38° and 67°. What is the third angle, and what type of triangle is it?

Flashcards

(13)

What is an interior angle?

The angle inside a shape between two of its edges. A triangle has three.

What do the interior angles of any triangle add up to?

180°, whatever the type of triangle.

What's the difference between a demonstration and a proof?

A demonstration shows a fact for the particular triangles you tried. A proof shows it for every triangle.

What's the first step of the parallel-lines proof of the angle sum?

Draw a line through one vertex, parallel to the opposite side.

Which two angle facts does the alternate-angles proof use?

Angles on a straight line add up to 180°, and alternate angles in parallel lines are equal.

What does ∴ mean?

Therefore.

Corresponding angles in parallel lines are…

Equal.

Co-interior angles in parallel lines…

Add up to 180°.

Vertically opposite angles are…

Equal.

How do you find the base angles of an isosceles triangle from its top angle?

Write 2m + (top angle) = 180° and solve for m.

The angles of a triangle are in the ratio 2 : 3 : 4. What are they?

9 parts, so each part is 20°: the angles are 40°, 60° and 80°.

Can a triangle have two obtuse angles?

No. Two obtuse angles already add up to more than 180°.

When can you say a triangle is isosceles?

When marks show two equal sides or angles, you're told it is, or the angle facts give two equal angles. Never just because it looks it.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Four triangles, one method

What type of triangle is it? Is it even a triangle?

Jay's triangle

One wrong angle fact

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