Stand still and turn to face the door. You just made an angle — even though nothing moved through space, your direction changed. That is the whole idea: an angle measures turning, not distance. Once you see it that way, the rules that follow stop being a list to memorise and start feeling obvious.
What an angle actually is
Picture two rays — two straight arms — that start from the same point. That shared point is the vertex, and the angle is the amount of turn it takes to swing one arm onto the other.
Notice what an angle does not depend on: how long you draw the arms. A turn of a quarter is a quarter whether the arms are an inch long or stretch across the room. Angle is about opening, never length. Lengthen an arm and the picture grows; the angle stays exactly the same.
The degree: slicing a full turn
To put a number on a turn, we slice one complete spin — all the way around, back to where you started — into 360 equal pieces. Each slice is one degree, written \(1^\circ\). So a full turn is \(360^\circ\).
That makes the useful landmarks easy to picture:
- A half-turn (a straight line) is \(180^\circ\).
- A quarter-turn (a square corner) is \(90^\circ\).
- A tiny nudge off straight-ahead might be only \(5^\circ\) or \(10^\circ\).
Why 360 and not 100? The number is ancient — the Babylonians liked it because it divides cleanly so many ways (by 2, 3, 4, 5, 6, 8, 9, 10, 12…). That convenience stuck for four thousand years, and we still ride on it.
Drag the arm below to feel the connection between the picture and the number. Watch where the value crosses \(90^\circ\) and \(180^\circ\) — those are the boundaries between the types you are about to meet.
The five types of angle
Every angle gets a name from how its size compares to those two landmarks, \(90^\circ\) and \(180^\circ\):
- Acute — less than \(90^\circ\). A sharp, narrow opening, like the hands of a clock at 1 o'clock.
- Right — exactly \(90^\circ\). A perfect square corner; we mark it with a small box instead of an arc.
- Obtuse — between \(90^\circ\) and \(180^\circ\). Wider than a corner but not yet flat.
- Straight — exactly \(180^\circ\). The two arms point in opposite directions and form a single line.
- Reflex — more than \(180^\circ\) (but less than \(360^\circ\)). You have swung past flat and are heading back around the long way.
In words The names are just regions on the dial. Below \(90^\circ\) it's acute, dead-on \(90^\circ\) it's right, between \(90^\circ\) and \(180^\circ\) it's obtuse, dead-on \(180^\circ\) it's straight, and beyond \(180^\circ\) it's reflex. Learn the two landmarks and the names sort themselves out.
Angles that team up: complementary and supplementary
Angles love to come in pairs that add up to one of our landmarks, and these two pairings appear everywhere.
Two angles are complementary when they add to \(90^\circ\) — together they fill a right angle. Two angles are supplementary when they add to \(180^\circ\) — together they fill a straight line.
The most common place you meet a supplementary pair is on a straight line: if a ray lands on a line, the two angles on one side must add to \(180^\circ\), because together they sweep the whole half-turn.
- Complementary angles add to \(90^\circ\), so the missing angle is whatever is left over from \(90^\circ\).
- Subtract the angle you know: \(90^\circ - 35^\circ = 55^\circ\).
- Check: \(35^\circ + 55^\circ = 90^\circ\). It fills a right angle, so it fits.
The complement of \(35^\circ\) is \(55^\circ\).
- Angles on a straight line are supplementary — together they fill the half-turn of \(180^\circ\).
- Subtract the known angle: \(180^\circ - 110^\circ = 70^\circ\).
- Check: \(110^\circ + 70^\circ = 180^\circ\). The pair lies flat along the line, exactly as it should.
The other angle is \(70^\circ\).
Tip — keep the two words straight "Complementary" comes first alphabetically (c before s) and goes with the smaller number, \(90^\circ\). A right angle "completes" a corner; a supplementary pair "supplies" the rest of a straight line.
When two lines cross: vertical angles
Let two straight lines slice through each other. They carve the space around the crossing point into four angles. Here is the beautiful part: the angles directly across from each other are always equal. These opposite pairs are called vertical angles (sometimes "vertically opposite").
Why must they match? Each pair sits on the same straight line, so each adds to \(180^\circ\). If one angle is \(70^\circ\), its neighbour on either side must be \(180^\circ - 70^\circ = 110^\circ\). And the angle opposite the \(70^\circ\) shares that same \(110^\circ\) neighbour, so it is forced to be \(70^\circ\) too. The two equal pairs are no coincidence — straightness guarantees them.
In words When two lines cross, opposite angles are equal and neighbouring angles add to \(180^\circ\). Know one of the four and you know all four.
Parallel lines and a transversal
Now take two parallel lines — railway tracks that never meet — and draw a single line straight across both. That crossing line is called a transversal, and because the two tracks point in exactly the same direction, the angles it makes repeat in a tidy pattern.
There are three relationships worth knowing by name:
- Corresponding angles are equal. Angles in the matching position at each crossing — same corner, top-left at one, top-left at the other — are identical. (Look for an F shape.)
- Alternate angles are equal. Angles on opposite sides of the transversal, tucked between the two parallel lines, match. (Look for a Z shape.)
- Co-interior (allied) angles add to \(180^\circ\). Angles on the same side of the transversal, both between the parallel lines, are supplementary. (Look for a C or U shape.)
The F, Z and C shapes are real lifesavers — once a pair of angles is sitting inside one of those letters, you instantly know whether they are equal or add to \(180^\circ\).
- The two angles sit on the same side of the transversal, both between the parallel lines — that is the co-interior (C-shape) pair.
- Co-interior angles are supplementary, so they add to \(180^\circ\).
- Subtract: \(180^\circ - 65^\circ = 115^\circ\).
The co-interior angle is \(115^\circ\).
Tip — these rules need parallel lines Corresponding, alternate and co-interior facts only hold when the two lines really are parallel. If the lines aren't parallel, the angles drift out of step and none of the equalities apply. Always check for the little arrowheads that mark parallel lines first.
Practice
Try each one yourself, then reveal the full solution.
1. What is the complement of \(35^\circ\)?
Complementary angles add to \(90^\circ\), so subtract the angle you have from \(90^\circ\):
\[ 90^\circ - 35^\circ = 55^\circ \]
Check: \(35^\circ + 55^\circ = 90^\circ\). The complement is \(55^\circ\).
2. Two angles lie on a straight line. One of them is \(110^\circ\). Find the other.
Angles on a straight line are supplementary — they add to \(180^\circ\):
\[ 180^\circ - 110^\circ = 70^\circ \]
Check: \(110^\circ + 70^\circ = 180^\circ\). The other angle is \(70^\circ\).
3. Two straight lines cross. One of the four angles is \(70^\circ\). Give the other three.
The angle opposite the \(70^\circ\) is its vertical angle, so it is also \(70^\circ\).
Each of the two neighbouring angles sits on a straight line with the \(70^\circ\), so it is supplementary:
\[ 180^\circ - 70^\circ = 110^\circ \]
That neighbour's own opposite is \(110^\circ\) as well. So the four angles are \(70^\circ\), \(110^\circ\), \(70^\circ\), \(110^\circ\) around the point.
The other three angles are \(110^\circ\), \(70^\circ\) and \(110^\circ\).