Parallel lines and a transversal

Drag the handle at the top of the slanted line (or use the slider) to change its angle. The eight angles it makes with the two parallel lines are numbered and coloured, and there are only ever two sizes: every blue angle equals every other blue angle, every orange equals every other orange, and a blue plus an orange makes 180°. Then switch to the triangle and drag its corners: the line through the top corner is parallel to the base, and the three angles of the triangle reappear side by side along it.

eight angles triangle sum angle 60°

What's going on

Two facts do all the work. At any crossing, angles next to each other along a straight line add to 180°, and angles opposite each other are equal. That is why a single crossing has only two sizes of angle. The parallel lines add a third fact: the slanted line meets both of them at the same tilt, so the four angles at the upper crossing are a copy of the four at the lower crossing. Corresponding angles (same corner at each crossing) are equal, alternate angles (the Z shape between the lines) are equal, and co-interior angles (the C shape) add to 180°.

Now the triangle. Draw the line through the top corner parallel to the base. The angle at the bottom-left corner and the blue angle at the top are alternate angles, so they are equal. Same for the bottom-right corner and the orange angle. The three angles at the top corner, blue, apex and orange, sit along a straight line, so they add to 180°. But those are the three angles of the triangle. So the angles of every triangle add to 180°, and the proof is the same whatever triangle you drag out.

Try to drag the corners so the sum is not 180°. You cannot, and the picture tells you why.

Sheets: Why a triangle's angles add to 180°.