A straight line is a half turn
A straight line through a point makes a half turn, 180°. Draw a ray from a point on the line, and it splits that half turn into two angles side by side, one on each side of the ray.
Suppose one of the two angles is 130°. The other angle is whatever is left of the half turn.
Angles on a line add to 180°
The two angles have no gap between them and do not overlap, so together they make the whole straight line, 180°. So the other angle is 180 − 130 = 50°. Check: 130 + 50 = 180.
The same is true however the ray leans. Angles on a straight line add up to 180°. To find a missing one, subtract the angle you know from 180.
90° + 90° = 180°: the two angles on a straight line make a half turn
Turn the arm to 130°
The arm opens upright, with 90° on each side of it. Turn it to 130°: the angle beside it shrinks to 50°. Wherever the arm points, the two angles add to 180°.
Why 180°
The fact follows from angles at a point. All the way around a point is a full turn, 360°. A straight line through the point cuts that turn into two equal halves, one above the line and one below it, and each half is 180°. The angles on one side of the line fill one half, so they add to 180°.
More than two angles
Two or more rays can meet at the same point on one side of a line. The angles between them still fill the half turn, so they still add to 180°. If three angles on a line are 45°, 70° and one unknown, the unknown one is 180 − 45 − 70 = 65°.
The rule also works with a letter. Two angles on a line are x° and 2x°, so x + 2x = 180. That gives 3x = 180 and x = 60, so the angles are 60° and 120°. Check: 60 + 120 = 180.
The usual mistakes
Subtracting from 360. A straight line is a half turn, 180°, not a full turn. The angle beside 130° on a line is 180 − 130 = 50°, not 360 − 130 = 230°.
Assuming the two angles are equal. They are equal only when both are 90°. Beside an angle of 130°, the other angle is 50°.
Using the rule where there is no straight line. The angles must sit on one side of a straight line, at a point on that line. If the two outer arms do not make a straight line, their angles need not add to 180°.
Naming an angle with three letters
When rays meet at a point O, an angle is named by three letters. ∠AOC is the angle at O between the rays OA and OC; the middle letter is always the point where the arms meet. "AOB is a straight line" means that A, O and B lie on one straight line, with O between A and B.
Worked example: Angles on a Straight Line in a Ratio
Question AOB is a straight line. Rays OC and OD split the angle above it so that ∠ AOC : ∠ COD : ∠ DOB = 2 : 3 : 4. Find ∠ COD and ∠ AOD.
1.AOB is a straight line, so the three angles along it add to 180°.
Three angles on one side of the straight line AOB, in the ratio 2 : 3 : 4. 2.The ratio 2 : 3 : 4 makes 2 + 3 + 4 = 9 units.
2 + 3 + 4 = 9 units fill the 180° of the straight line. 3.9 units = 180°, so 1 unit = 180° ÷ 9 = 20°.
One unit is 180° ÷ 9 = 20°. 4.∠ COD = 3 units = 3 × 20° = 60°.
∠ COD is 3 units: 60°. 5.∠ AOD is ∠ AOC and ∠ COD together: 2 + 3 = 5 units.
∠ AOD reaches from OA past OC to OD: 5 units. 6.∠ AOD = 5 × 20° = 100°.
∠ AOD = 5 × 20° = 100°.
Answer: ∠ COD = 60°; ∠ AOD = 100°
Common mistakes
- Sharing 360° among the units instead of 180°: rays on one side of a straight line make half a turn, not a whole one.
- Reading ∠ AOD as the single part ∠ DOB (4 units) instead of the two parts from OA round to OD.
Cutting an angle in half
To bisect an angle is to cut it into two equal halves, and the ray that does it is called the angle's bisector. The bisector of an angle of 80° splits it into two angles of 80 ÷ 2 = 40°.
Worked example: Bisectors of Two Angles on a Straight Line
Question AOB is a straight line and ∠ AOC = 70°. OD bisects ∠ COB and OE bisects ∠ AOC. Find ∠ DOE.
1.AOB is a straight line, so ∠ AOC + ∠ COB = 180° and ∠ COB = 180° − 70° = 110°.
On the straight line AOB, ∠ COB = 180° − 70° = 110°. 2.OD bisects ∠ COB: ∠ COD = 110° ÷ 2 = 55°.
OD bisects ∠ COB: ∠ COD = 55°. 3.OE bisects ∠ AOC: ∠ EOC = 70° ÷ 2 = 35°.
OE bisects ∠ AOC: ∠ EOC = 35°. 4.∠ DOE is the two halves next to OC: ∠ EOC + ∠ COD = 35° + 55° = 90°.
∠ DOE = 35° + 55° = 90°. 5.In general: half of ∠ AOC plus half of ∠ COB is half of 180°.
Half of ∠ AOC plus half of ∠ COB is half of 180°: the bisectors are always perpendicular. 6.So the bisectors of two angles on a straight line are always perpendicular, whatever ∠ AOC is.
Answer: ∠ DOE = 90°
Common mistakes
- Halving 70° and stopping: ∠ DOE needs the half of ∠ COB as well.
- Drawing OD inside ∠ AOC: OD bisects ∠ COB, on the other side of OC from OE.