Theorem

Angles on a straight line and vertically opposite angles

Geometry • Line • Angles at a Point

Statement

Angles on one side of a straight line at a point on it sum to

18 0^{\circ}

, all the angles round a point sum to

36 0^{\circ}

, and where two straight lines cross, each pair of vertically opposite angles is equal.

How to use it

1When several rays leave a point of a straight line on one side of it, set the angles they make adding to $18 0^{\circ}$ and solve for the unknown one.
2At a crossing, carry a known angle straight across to its vertically opposite twin; the two angles adjacent to it are its supplements.
3Use the $36 0^{\circ}$ sum when the rays surround the point rather than lying to one side of a line.

Worked example

X

O

Y

are collinear and rays

O A

O B

leave

O

on one side of

X Y

, with

∠ X O A = 4 0^{\circ}

and

∠ A O B = 7 5^{\circ}

. Find

∠ B O Y

, and the angle vertically opposite

∠ X O A

formed when

A O

is produced beyond

O

Step-by-step

1
$∠ X O A + ∠ A O B + ∠ B O Y = 18 0^{\circ}$
Why: The three angles lie on one side of the straight line $X Y$ .
2
$∠ B O Y = 18 0^{\circ} - 4 0^{\circ} - 7 5^{\circ} = 6 5^{\circ}$
3
the angle vertically opposite $∠ X O A$ is $4 0^{\circ}$
Why: Producing $A O$ beyond $O$ makes two crossing lines, and vertically opposite angles are equal.

Answer

∠ B O Y = 6 5^{\circ}

, and the vertically opposite angle is

4 0^{\circ}

Watch out

The

18 0^{\circ}

sum needs the three points to be genuinely collinear — check the figure marks it rather than assuming it from the drawing. Vertically opposite angles are equal at ANY crossing, parallel lines or not; the ADJACENT angles at a crossing are supplementary, not equal.

Also known as

angles on a straight line, linear pair, vertically opposite angles, angles at a point, supplementary adjacent angles, angles round a point

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (I)

2019 (II)

2018 (II)

2018 (I)

2017 (II)

2017 (I)