Angle Chasing

Two parallel lines with a bent line between them is a favourite competition picture. The trick is to draw a third line through the bend, parallel to the other two. Now the angle at the bend splits into two parts. The upper part is alternate to the forty degree angle at A, so it is forty degrees. The lower part is alternate to the sixty five degree angle at C, so it is sixty five degrees. So x is forty plus sixty five, which is one hundred and five degrees. Whenever a line bends between parallels like this, the bend angle is the sum of the two angles it makes with the parallel lines.

In triangle A B C the ticks show that A B equals A C, so the angles opposite those sides, at B and at C, are equal. The angle at A is forty degrees, so the other two share one hundred and forty degrees: seventy each. Now extend B C to D. Angle A C D sits on a straight line with the seventy degree angle, so it is one hundred and ten degrees. There is a shortcut called the exterior angle theorem: an exterior angle of a triangle equals the sum of the two interior angles opposite it. Forty plus seventy is one hundred and ten. In competition problems you will use this shortcut again and again.

Pick one corner of a polygon and draw every diagonal from it. A hexagon splits into four triangles, so its interior angles add to four times one hundred and eighty, which is seven hundred and twenty degrees. In general, a polygon with n sides splits into n minus two triangles, so its interior angles add to n minus two, times one hundred and eighty. The exterior angles are even easier: walk all the way round any polygon and you turn through one full turn, three hundred and sixty degrees. So each exterior angle of a regular octagon is three hundred and sixty divided by eight, forty five degrees, and each interior angle is one hundred and eighty minus forty five, one hundred and thirty five. Working backwards, if each exterior angle is twenty four degrees, the polygon has three hundred and sixty divided by twenty four, which is fifteen sides.

Here is a five-pointed star whose points are not all the same. The surprising fact is that its five tip angles always add to one hundred and eighty degrees. Here is why. Look at the shaded triangle with tip A at the top. Each of its two bottom corners is an exterior angle of a small triangle, so each one equals the sum of two other tips. That means the three angles of the shaded triangle hold all five tips between them, and the angles of a triangle add to one hundred and eighty. So thirty two plus thirty six plus forty one plus forty five plus x is one hundred and eighty. The four known tips make one hundred and fifty four, so x is twenty six degrees.

Regular polygons fit together around a point with no gap only if their interior angles add to exactly three hundred and sixty degrees. A square has ninety degrees. A regular octagon has one hundred and thirty five. Ninety plus one hundred and thirty five plus one hundred and thirty five is three hundred and sixty, so a square and two octagons fit perfectly. That is the pattern on many tiled floors. To find a missing polygon, work out the gap it has to fill, turn that interior angle into an exterior angle, and divide it into three hundred and sixty. Quiz and exercises now.