This is Lesson 3, Florence. You already have angles on a straight line, angles around a point, vertically opposite angles, and the three parallel-line pairs. Today's lesson is about the triangle — the shape all those rules were quietly building towards.
There are three things to learn: the rule that ties every triangle together, two special triangles worth knowing by name, and one neat shortcut.
It works as before — small steps, answers tapped or typed, a quiet nudge if one isn't right, and a part you can drag with your finger. Tap Continue when you're ready.
Every triangle, however it is stretched or squashed, has three angles inside it — one at each corner. And here is the useful part: those three angles always add up to 180°. A tall thin triangle, a wide flat one — it makes no difference at all.
So if you know two of the angles, you can always find the third: add the two you have, then take the total away from 180°.
See it for yourself. Drag any corner of the triangle below and watch the three angles. Each one changes — but the total underneath never leaves 180°.
The three angles of any triangle add up to 180°. There are three here, so together they make 180°.
52° + 81° + x = 180°
Add the two you know, then take the total away from 180°.
52° + 81° = 133°, and 180° − 133° = 47°
The three angles of a triangle always add up to 180°.
Add the two you know: 90° + 35° = 125°.
Then 180° − 125° = 55°.
The three angles add up to 180°.
48° + 61° = 109°.
Then 180° − 109° = 71°.
The three angles add up to 180°.
25° + 115° = 140°.
Then 180° − 140° = 40°.
Some triangles are special, and being able to name them saves you work.
An isosceles triangle has two sides the same length. Wherever two sides match, the two angles opposite them match as well — so an isosceles triangle has two equal angles, often called the base angles. The small dashes on the diagram mark the equal sides.
An equilateral triangle goes further: all three sides are equal, so all three angles are equal too. Since they must still add to 180°, each one is 180° ÷ 3 = 60°.
All three angles still add up to 180°. Take the apex angle away first to see what is left for the two base angles.
180° − 36° = 144°
That 144° is shared equally between the two base angles, because they are equal. So halve it.
x = 144° ÷ 2 = 72°
All three angles add up to 180°.
The two base angles together are 70° + 70° = 140°.
So the apex is 180° − 140° = 40°.
All three angles add up to 180°.
Take away the apex: 180° − 36° = 144°.
The two equal angles share that: 144° ÷ 2 = 72°.
An equilateral triangle has three equal angles.
They add up to 180°, so each is 180° ÷ 3 = 60°.
If you take one side of a triangle and carry it on past a corner, the angle that opens up outside the triangle is called an exterior angle.
There is a neat fact about it. An exterior angle is always equal to the two interior angles it is not next to — the two at the far corners — added together.
It saves a step: instead of finding the inside angle first and then subtracting from 180°, you can jump straight to the exterior angle by adding the two opposite ones.
The exterior angle equals the two interior angles at the far corners, added together. Those are the 64° and the 58°.
x = 64° + 58°
So just add them.
x = 64° + 58° = 122°
An exterior angle equals the sum of the two opposite interior angles.
So it is 50° + 65° = 115°.
The exterior angle equals the two opposite interior angles added together.
So those two angles add to 130°. One is 70°, so the other is 130° − 70° = 60°.
This last part has two steps and uses two of today's ideas at once. Here is the first.
The three angles add up to 180°.
90° + 34° = 124°, then 180° − 124° = 56°.
You found that c = 56°. The triangle's three angles are 90°, 34° and 56°. Now use the exterior-angle idea.
The exterior angle equals the two opposite interior angles added together.
Those are 90° and 34°, so the exterior angle is 90° + 34° = 124°.
That's Lesson 3 done, Florence.
You can now find any missing angle in a triangle: the three always make 180°, an isosceles triangle hands you two equal angles, an equilateral one is 60° all round, and the exterior-angle shortcut saves you a step. That is real GCSE groundwork — the same facts sit underneath much harder questions later on.
Three lessons in, and the week is nearly done. One more after this — whenever you're ready for it.
You can close this page now — or step back through any part you'd like to see again.