Angles: Naming, Measuring and Adding

First, the pieces

Name	What it is
point	a single position
line	straight, going on forever both ways
ray	straight, starting at a point and going on forever one way
line segment	straight, with two endpoints

An angle is formed where two rays meet at a shared endpoint. That shared point is the vertex, and the rays are the arms.

An angle measures turn, not length

This is the point people miss.

An angle is how far one arm is turned from the other. The length of the arms makes no difference at all. Drawing them twice as long leaves the angle exactly as it was.

That is why a small drawing and a large one can show the same angle, and why you cannot judge an angle by how big the picture is.

Why a degree is a 360th of a turn

Picture a circle with the vertex at its center. A full turn all the way round is divided into 360 equal parts, and each part is one degree.

$$1° = \tfrac{1}{360} \text{ of a full turn}$$

So an angle of $n°$ turns through $n$ of those parts.

Turn	Degrees
full turn	$360°$
half turn	$180°$
quarter turn	$90°$
three quarter turn	$270°$

Why 360? It is a convention inherited from ancient astronomy, and it has stuck because $360$ divides evenly by so many numbers. Among them are $2, 3, 4, 5, 6, 8, 9, 10$ and $12$. That keeps the common fractions of a turn whole, so a third of a turn is exactly $120°$ rather than something awkward.

Naming angles by size

Name	Measure
acute	less than $90°$
right	exactly $90°$
obtuse	more than $90°$, less than $180°$
straight	exactly $180°$
reflex	more than $180°$

A right angle is the corner of a square, and it is usually marked with a small square rather than an arc.

Two lines meeting at a right angle are perpendicular. Two lines that never meet, staying the same distance apart forever, are parallel.

Measuring with a protractor

Place the protractor’s center on the vertex and line one arm up with the zero. Read where the other arm crosses the scale.

A protractor has two scales, running opposite ways, and choosing the wrong one is the usual error. The check is the angle’s name:

• an angle that looks acute must read under $90$
• an angle that looks obtuse must read over $90$

If your reading disagrees with what you can see, you are on the wrong scale. Reading $130$ for an angle you can see is acute means the answer is $50$.

Angle measures add

Put two angles side by side, sharing an arm. Together they make one larger angle, and its measure is the sum.

$$30° + 45° = 75°$$

The reason is what an angle measures. Turning through $30°$ and then a further $45°$ from there is one turn of $75°$. The turns accumulate.

That works backwards too. If a $75°$ angle is split into two, and one part is $30°$, the other is:

$$75 - 30 = 45°$$

This is what lets you find a missing angle from the ones around it, and it is the basis of every angle relationship that comes later.

Worked examples

Common mistakes

Practice problems