Angles, Euclid and the number 360

In the site’s Angle Puzzles, you find the angle marked “?” without a protractor. All you use are properties of shapes known for a very long time. This article looks at a few of them, and at the convention called the degree.

The properties the puzzles use (the site’s “Angle Puzzles”)

The puzzles number 100 in all, in seven levels: lines and triangles, equal sides and exterior angles, parallel lines and bends, polygons, circles, combined ideas and challenges. For every problem, this site’s program checked that each angle in the drawing matches its stated value within half a degree (angle-puzzle/engine.test.ts). If you get stuck, open the hints one step at a time.

Vertical angles are equal

When two straight lines cross, the angles facing each other (vertical angles) are equal. This is Proposition 15 of Book I of Euclid’s Elements. It is the first property you reach for, and it keeps coming back in later problems.

The exterior angle of a triangle

Extend one side of a triangle, and the exterior angle formed is larger than either of the two interior angles not next to it. That is Proposition 16 of Book I. The stronger form, “the exterior angle equals the sum of the two interior angles not next to it”, is shown in Proposition 32. Proposition 32 is also where the angles of a triangle are shown to add up to two right angles, that is, 180 degrees.

Parallel lines and angles

When a line crosses two parallel lines, the alternate angles and corresponding angles are equal. Conversely, if they are equal, the two lines are parallel. This is used a lot in the level “Parallel lines and bends”. Parallel lines hold a special place in the Elements. The long search for a geometry that does without the “parallel postulate” is said to have led to non-Euclidean geometry.

The angles of a polygon

Draw diagonals from one corner of an n-sided polygon and it splits into (n − 2) triangles. So the interior angles add up to (n − 2) × 180 degrees: 360 for a quadrilateral, 540 for a pentagon. This is sound mathematics. But it cannot be said that Euclid himself proved it in the Elements.

The convention of 360 degrees

The Elements has no unit called the degree. Euclid compared angles as multiples of a right angle. Geometry existed before anyone used degrees.

So why divide a circle into 360? The habit is said to come from Babylonian astronomy. Hipparchus (second century BC, Rhodes) is said to have brought the 360 division into Greek astronomy. But the reason is not known. There are several theories. One is that the year is about 360 days long and the Sun moves about one degree a day. You often see the explanation “the Babylonians used base 60”, but some sources say the link is weak, so it is not decisive.

Why it is convenient is clear. 360 has many divisors, 24 of them (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360). So splitting a circle into 2, 3, 4, 5, 6, 8, 10 or 12 equal parts gives whole numbers of degrees. But that does not show it became 360 because it was convenient. Why it is handy and where it came from are separate stories.

Ways to think

• First look for familiar shapes in the figure: lines, triangles, parallels, circles.
• From the angles you know, add and subtract your way toward “?”.
• With no clue, think of a helper line. For a polygon, try splitting it into triangles.

Hints in Angle Puzzles come in the order of idea, strategy, then the exact next step. Try looking at just the idea first.