Why angles behave: proofs, parallels and polygons

In earlier layers you learned angle facts, tested them, and saw them work every time. This layer asks the deeper question: why must they be true? And it pushes further, into parallel lines, triangles and polygons, where those simple facts combine into powerful results.

You will learn to write a short proof: a chain of statements, each backed by a reason, starting from things everyone agrees on. This is how mathematics has worked for more than 2,000 years, since Greek geometers collected their proofs in Euclid's Elements, and it is still how mathematicians convince each other today.

Along the way: why a full turn is 360°, what makes railway tracks parallel, why every triangle's angles add to 180° (and a surprising place where they do not), and how to solve missing-angle problems that would stump most adults.

Nobody proved that a full turn is 360°. It is a choice, a unit agreed by people, like choosing 100 paise in a rupee. So why this odd number?

The most common explanation goes back to astronomers in ancient Babylon (in today's Iraq), around 4,000 to 2,000 years ago:

• They counted in base 60 (sexagesimal), not base 10. We still see it in our clocks: 60 seconds in a minute, 60 minutes in an hour. Each degree is also split into 60 minutes (′) and each minute into 60 seconds (″).
• A year has about 365 days, and the Sun appears to move once round the sky in a year. 360 is close to 365, so a degree is roughly the Sun's movement against the stars in one day.
• 360 has a huge number of 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). A full turn can be split into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths with whole-number answers. Compare 100, which has only 9 divisors.

Historians still debate exactly how the choice was made, but the usefulness of 360 is not in doubt.

In Understand you saw an argument that vertically opposite angles are equal. Now we set it out as a formal proof, the way it appears in Euclid's Elements, Book I, Proposition 15. The ancient writer Eudemus credited the discovery to Thales of Miletus, around 600 BCE.

Every proof has three parts:

• Given: what we are told.
• To prove: what we want to show.
• Proof: statements, each with a reason. Reasons can only be definitions, given facts, earlier proved results or basic agreed facts (axioms).

Once one fact is proved, it becomes a tool for proving the next. Here are two more short proofs that use only linear pairs.

Angles around a point add to 360°. Draw any line through the point. The angles on one side of the line make a straight angle (180°) and the angles on the other side make another straight angle (180°). Together: 180° + 180° = 360°. (If a ray does not lie along the line, it simply splits one of those straight angles into parts that still add to 180°.)

Supplements of equal angles are equal. If ∠P = ∠Q, then 180° − ∠P = 180° − ∠Q. This small fact is used again and again, for example to show that if one angle at a crossing is 90°, all four are 90°.

Parallel lines are lines in the same flat surface that never meet, however far they are extended, like the two rails of a railway track or the lines on a ruled notebook page. We write l ∥ m.

A line that crosses two (or more) other lines is called a transversal. When a transversal cuts two lines, it makes eight angles: four at each crossing. Number them 1 to 4 at the top crossing and 5 to 8 at the bottom crossing, in the same positions (top-left, top-right, bottom-right, bottom-left).

• Angles between the two lines are interior angles (3, 4, 5, 6 in the usual picture).
• Angles outside the two lines are exterior angles (1, 2, 7, 8).

The pairs formed get special names, whether or not the lines are parallel. But the magic happens when the lines are parallel.

Co-interior angles add to 180°. Take ∠4 and ∠5, which are both interior and on the same side of the transversal. ∠4 and ∠3 form a linear pair at the top crossing, so ∠4 + ∠3 = 180°. We just proved ∠3 = ∠5. Replace ∠3 with ∠5: ∠4 + ∠5 = 180°. ∎

So when the lines are parallel, the eight angles come in only two sizes: four equal "small" angles and four equal "large" angles, and small + large = 180° (unless the transversal is perpendicular, when all eight are 90°).

The facts about transversals also work backwards. These backwards statements are called converses:

• If a transversal makes equal corresponding angles with two lines, the lines are parallel.
• If it makes equal alternate interior angles, the lines are parallel.
• If it makes co-interior angles that add to 180°, the lines are parallel.

This is how carpenters, engineers and surveyors check parallel lines in real life. A railway track inspector cannot walk to infinity to check that the rails never meet, but can check that a straight sleeper (a transversal) meets both rails at equal angles. A carpenter uses a set-square sliding along a straight edge to draw parallel lines, because every line drawn meets the edge at the same angle.

In Investigate, torn paper corners suggested that a triangle's three angles make a straight line. Here is the proof, essentially Euclid's Proposition I.32.

Given: triangle ABC with angles ∠A, ∠B and ∠C at its corners. To prove: ∠A + ∠B + ∠C = 180°.

1. Through vertex A, draw line PQ parallel to BC. (Through a point not on a line there is exactly one parallel line.)
2. Along PQ at A there are three angles side by side: ∠PAB, ∠BAC and ∠CAQ. They make a straight line, so ∠PAB + ∠BAC + ∠CAQ = 180° (angles on a straight line).
3. ∠PAB = ∠B, because they are alternate interior angles (PQ ∥ BC, transversal AB).
4. ∠CAQ = ∠C, because they are alternate interior angles (PQ ∥ BC, transversal AC).
5. Substitute into step 2: ∠B + ∠A + ∠C = 180°. ∎

This is exactly the tear-the-corners experiment in disguise: the parallel line at A is where the torn corners ∠B and ∠C slide to.

A quadrilateral (four-sided shape) can always be cut by a diagonal into two triangles. The angles of the two triangles together make up exactly the four corner angles of the quadrilateral. So:

Angle sum of a quadrilateral = 2 × 180° = 360°.

This works for squares, rectangles, kites, trapeziums and lopsided shapes alike, as long as the shape is not crossed over itself. (For a shape with a “dent” in it, choose the diagonal from the dented corner, which lies inside the shape.)

The same trick works for any polygon. From one corner, draw diagonals to all the non-neighbouring corners. A polygon with n sides splits into n − 2 triangles, so:

Angle sum of an n-sided polygon = (n − 2) × 180°.

A classic puzzle: two parallel lines, and a path that goes from one to the other with a bend in it, pointing between the lines (like an arrowhead >). How big is the angle at the bend?

Trick: draw a third parallel line through the bend. It splits the bend angle into two parts. The top part is an alternate angle with the angle at the top line, and the bottom part is an alternate angle with the angle at the bottom line. So:

angle at the bend = (angle at top line) + (angle at bottom line).

This trick, adding an extra line to create angles you know, is called a construction in a proof. It is the same idea we used for the triangle sum. Good problem-solvers are always asking: what extra line would help?

Deep understanding means knowing where the rules stop working. A few edge cases:

• Reflex angles and naming. ∠ABC normally means the smaller angle. If the reflex angle is meant, say “reflex ∠ABC”.
• Zero-width “triangles”. If three points lie on one line, there is no triangle: two angles are 0° and the third is 180°. The angle sum is still 180°, a hint that the rule is very robust.
• Parallel lines in a triangle problem. The triangle proof needs a parallel line; on a sphere there are no parallel lines, which is exactly why the sum changes there.
• Transversal at 90°. All eight angles are right angles. Corresponding, alternate and co-interior facts still hold, but they all look the same.
• More than 360°. A turn can be 450° (a full turn plus a quarter turn). As a direction, it ends in the same place as 90°. Wheels, fans and spinning bowlers can turn through thousands of degrees.