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 $\alpha = \gamma$.

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 $180^\circ$.

Exercises

Type the number of degrees alone, e.g. 115.

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.