Space
Proving angle facts
Use the linear-pair axiom to prove vertically opposite angles equal, and apply the angle-sum and exterior-angle theorems.
You have used angle facts for years — angles on a line make 180°, vertically opposite angles are equal, a triangle's angles come to 180°. This year the question changes from what is true to why it has to be. And the answer, nearly every time, is the same single starting point: the angles on one side of a straight line add up to 180°. Everything else is squeezed out of that.
The one thing assumed
The linear-pair axiom: if a ray stands on a straight line, the two angles it makes add up to 180°. It is assumed, not proved. The facts about angles meeting at a line below are argued from it. The parallel-line facts need one more assumption of their own — the corresponding-angles axiom — which is exactly the kind of extra assumption the Euclid skill says could be chosen differently.
Worked example
Prove that vertically opposite angles are equal.
Two lines cross, making four angles. Go round them in turn and call them a, b, c and d.
Naming them is what lets the argument talk about all four at once, whatever their sizes.
Have a play
One side of this triangle has been carried straight on past the corner B. Find out what each of the four angles here is called.
Tap any part of the picture.
Try it together
Two angles of a triangle are 42° and 58°. Find the exterior angle at the third corner.
Two routes reach the same number here, which is the whole point of the theorem.
1.Add the two given angles: what is 42 + 58?
Watch out
The exterior angle theorem uses the two interior angles the exterior angle does not touch. The one it does touch is its linear-pair partner, and those two add to 180° instead of being equal.
Have a go
Have a go on your own: an exterior angle of a triangle is 125°, and one opposite interior angle is 60°. What is the other, in degrees?
Hint: The two opposite interior angles have to add up to the exterior one.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.