Angles

An angle is the figure made by two arms leaving a common point — the vertex. Its measure is given in degrees: a full turn is $360^\circ$, a quarter turn is $90^\circ$.

One thing to settle straight away: the measure of an angle depends only on how far its arms open, never on how long they are. The same angle drawn once with short arms and once with long arms has the same measure.

Kinds of angle

The full list of names:

Pairs of angles

Two pairs turn up in problems more than any others, because they let you work out an angle nobody measured.

Complementary angles together make a right angle:

Supplementary angles sit on either side of the same arm and together make a straight line:

Worth knowing too are vertical angles — the ones opposite each other where two lines cross. They are always equal, which follows directly from supplementary pairs: if $\alpha + \beta = 180^\circ$ and $\beta + \gamma = 180^\circ$, then $\alpha = \gamma$.

Angles at two parallel lines

Two parallel lines cut by a third one — a transversal — make eight angles, four at each crossing. Only one of them has to be measured: the other seven follow from it.

Corresponding angles sit at two different crossings but in the same position at each: on the same side of the transversal and on the same side of their own line. At parallel lines they are equal:

Alternate angles sit between the parallels, on opposite sides of the transversal. They are equal too:

Co-interior angles sit between the parallels on the same side of the transversal. They are the one pair of the three that is not equal but supplementary:

All three relations rest on the lines being parallel — for two lines that meet somewhere, none of them holds. The rule also works backwards, and that is how parallelism is checked in practice: if a pair of corresponding angles is equal, the lines are parallel.

The angle sum of a triangle

This is the single most important rule in plane geometry:

It is easy to see for yourself: cut a triangle out of paper and tear off its three corners. Laid side by side they always form a straight line, i.e. a straight angle.

The practical consequence: to know all three angles you only need to measure two.

Two facts worth keeping to hand follow from the rule:

• in an equilateral triangle every angle is $180^\circ : 3 = 60^\circ$;
• in a right triangle the right angle takes $90^\circ$, so the two acute angles are complementary — together they make $90^\circ$.

The rule also rules certain triangles out: there is no triangle with two right angles, and none with two obtuse ones — those two angles alone would already use up the whole $180^\circ$.

The angle sum of a polygon

A triangle is not an exception but the simplest case of a general rule. Draw every diagonal from one vertex of a polygon with $n$ sides: they cut it into $n - 2$ triangles, and each triangle contributes its own $180^\circ$.

For a triangle this gives $(3 - 2) \cdot 180^\circ = 180^\circ$, the rule we started from. For a quadrilateral it gives $(4 - 2) \cdot 180^\circ = 360^\circ$ — which a rectangle makes obvious: four right angles are exactly $360^\circ$.

In a regular polygon all the angles are equal, so one angle is the sum divided by how many there are:

For a regular hexagon that is $\tfrac{4 \cdot 180^\circ}{6} = 120^\circ$, for a regular pentagon $108^\circ$, and for a square $90^\circ$. Dividing by $n$ is allowed for a regular polygon only — in any polygon the sum is the same, but the individual angles need not be equal. Regular polygons themselves — their side and the radii of their inscribed and circumscribed circles — are a topic of their own: the circle.

That lesson also owns the angles whose vertex lies on a circle or at its centre: the inscribed and the central angle obey a relation of their own, one that none of the pairs here can produce.

Exercises

Type the number of degrees alone, e.g. 115 — the sum of a polygon's angles included. For parallel lines the question says which of the three pairs it is asking about.

Common mistakes

• Judging an angle by the length of its arms — the measure is the opening, not how far the arms reach.
• Confusing complementary with supplementary — complementary make $90^\circ$, supplementary $180^\circ$.
• Adding the angles "by eye" — a triangle always gives $180^\circ$, even when the drawing is inaccurate.
• Looking for a triangle with two right angles — no such triangle exists.
• Forgetting the right angle in a right triangle — the two acute angles share $90^\circ$, not $180^\circ$.
• Dropping the degree sign — $65$ and $65^\circ$ are two different things; a measure is written with its degree sign.
• Using the parallel-line rules on lines that are not parallel — equal corresponding and alternate angles start from parallelism.
• Confusing co-interior angles with alternate ones — the first pair adds up to $180^\circ$, the second is equal.
• Dividing the angle sum by $n$ for any polygon — a single angle of $\tfrac{(n - 2) \cdot 180^\circ}{n}$ belongs to a regular polygon only.
• Multiplying by $n$ instead of $n - 2$ — there are always two fewer triangles than sides.