Continue to Angle Relationships

Two ideas, taught in order: what happens when two straight lines cross, and what happens when several rays share one starting point.

Part 1

The Vertically Opposite Angles

Vertically opposite angles are two non-adjacent angles formed by the intersection of two straight lines.

Lines AB and CD intersect at M

In the figure, straight line AB meets straight line CD at the point M. Four angles are created. Two angles that sit across the vertex from each other — sharing only the point M and no ray — are vertically opposite.

• Angles ∠AMC and ∠BMD are vertically opposite (angles 1 and 3).
• Angles ∠AMD and ∠BMC are vertically opposite (angles 2 and 4).
• Neighbouring angles such as ∠AMC and ∠AMD are adjacent, not vertically opposite — together they form a straight angle of 180°.

Rule 1

The two vertically opposite angles are congruent (equal in measure):
m(∠AMD) = m(∠BMC) and m(∠AMC) = m(∠BMD)

Why is it true?

∠AMC and ∠AMD lie on the straight line CD, so their measures add to 180°. In the same way ∠AMD and ∠BMD lie on the straight line AB, so they also add to 180°. Both pairs share ∠AMD, therefore:

m(∠AMC) = 180° − m(∠AMD) = m(∠BMD)

Part 2

The Accumulative Angles at a Point

Rays that share one starting point split the full turn around that point into angles.

If MA, MB and MC are rays that have the same starting point M, then ∠AMB, ∠BMC and ∠CMA are called accumulative angles at the point M.

Rule 2

The sum of the measures of accumulative angles at a point is equal to 360°:
m(∠AMB) + m(∠BMC) + m(∠CMA) = 360°

This works for any number of angles — three rays, four rays or ten rays. One complete turn is always 360°, so a missing angle is found by subtracting all the known angles from 360°.

Useful partial sums

Straight angle = 180° (half a turn)
Right angle = 90° (quarter turn)
Full turn = 360° = 4 right angles

Three rays from M: the three angles total 360°

Part 3

Combining the two rules

Exam figures usually mix a straight line, a right angle and a pair of vertically opposite angles.

x = 40 (V.O.A.) and y = 140 (V.O.A.)

? = 360° − (90° + 150°) = 120°

70° + 2x = 180° → x = 55

Common mistakes to avoid

✗ Treating two adjacent angles as vertically opposite. They add to 180°, they are not equal (unless both are 90°).
✗ Using 180° instead of 360° when several rays surround a point.
✗ Forgetting to substitute back: after solving for x, check whether the question asked for x or for the angle measure.
✗ Ignoring the small square marker — it always means exactly 90°.

A reliable method

Mark every known measure on the figure.
Decide the relationship: straight line (180°), point (360°) or V.O.A. (equal).
Write one equation from that relationship.
Solve for the unknown and simplify.
Re-read the question and state exactly what was asked.

See the method used in examples →