The angle sum and the exterior angle

Tear the three corners off any paper triangle and lay them point to point. They fill a straight line, 180°, whatever triangle you started with.

Tear off the corners and line them up

Draw a triangle on scrap paper, any shape at all, and cut it out. Mark each corner with a dot so you will know which tip is which. Now tear the three corners off and place them side by side so that the three dots meet at one point and the edges touch. The torn pieces close up into a straight line along the bottom. Try it again with a long thin triangle and with a nearly equilateral one. The fit is the same each time. Three corners of a triangle together make a straight angle, 180°.

Tearing and dragging both point to the answer, but neither proves it. Paper tears unevenly, and a screen rounds each angle to a tenth of a degree, which the panel admits. A test on ten triangles, or a thousand, is still a test on some triangles. A proof has to work for every triangle at once, including ones nobody has drawn.

One parallel line turns the experiment into a proof

The proof in OpenStax, which goes back to Euclid, draws one extra line. Through the top corner C, draw the line parallel to the base AB. Now look at the three angles that sit along that new line at C. The left one and angle A are alternate angles between two parallel lines, so they are equal. The right one and angle B are alternate angles too, so they are equal. The middle one is angle C itself. Those three angles fill a straight line through C, so they add to 180°. Replace the outer two by A and B and you get A + B + C = 180°. Nothing in the argument depends on the shape of the triangle.

The exterior angle is the sum of the two far corners

Extend one side of a triangle past a corner. The angle between the extension and the next side is an exterior angle. At corner C, the exterior angle and the interior angle C sit side by side on a straight line, so exterior angle = 180° − C. But A + B is also 180° − C. Therefore the exterior angle at C equals A + B, the sum of the two interior angles that are not next to it. Euclid states this in the same Proposition 32, before the angle sum. If A = 40° and B = 65°, the exterior angle at C is 105°, and you never needed to find C, which is 75°.

Take one exterior angle at each corner and add them. Each exterior angle is 180° minus its interior angle, so the three together are 3 × 180° − 180° = 360°. A walker who goes once round a triangular park turns through one full turn. That is Proclus’s corollary for the triangle.

Finding missing angles with algebra

The angle sum turns many picture puzzles into one-line equations. OpenStax gives a triangle whose angles are x, x + 17 and 3x − 62 degrees. Adding them, 5x − 45 = 180, so 5x = 225 and x = 45. The three angles are 45°, 62° and 73°, and they do add to 180°. Always finish with that last check; a slip in the algebra shows up at once. For an isosceles triangle, remember that the two base angles are equal, so an apex angle of 40° leaves (180 − 40) ÷ 2 = 70° for each base angle.