Proving the angle sums of triangles and polygons

0:00Why do a triangle's angles always add to 180 degrees? Let's prove it.

0:06Call its interior angles, the angles inside at each corner, a, b and c.

0:14Through the top vertex, or corner, draw a line parallel to the base. Parallel lines never meet.

0:22A transversal, a line crossing two parallel lines, makes a Z shape. The alternate angles inside the Z are equal.

0:31The same rule copies angle c up to the top.

0:35Now b, a and c sit on a straight line, so they add to 180 degrees.

0:43So a plus b plus c is 180 degrees, in every triangle.

0:49A polygon is a closed flat shape with straight sides. This hexagon has 6.

0:56From one vertex, draw the diagonals, lines to each corner not next to it.

1:02That splits it into 1, 2, 3, 4 triangles.

1:07Each sits on a side away from that vertex, so n sides make n minus 2 triangles: 6 minus 2 is 4.

1:17Their angles fill the hexagon's corners, so the angle sum, the total of its angles, is 4 times 180, or 720 degrees.

1:28A regular hexagon has equal sides and equal angles, so each angle is 720 divided by 6, or 120 degrees.

1:39A common mistake: split from the centre into 6 triangles, and get 6 times 180, or 1080 degrees.

1:49But the angles around the centre make a full turn, 360 degrees, and are not angles of the hexagon.

1:57Take them away: 1080 minus 360 is 720 degrees, the same answer.

2:05Recap: a polygon with n sides has an angle sum of n minus 2, times 180 degrees.