Angles at a Point and Vertically Opposite Angles
Angles that fill a full turn around a point add to \(360^\circ\), and when two straight lines cross, the opposite angles are equal. In NSW Year 7 maths (NESA outcome MA4‑ANG‑C‑01) these are called angles at a point and vertically opposite angles.
You will find missing angles by subtracting from \(360^\circ\), combine equal opposite angles with angles on a straight line, and give a reason with each step of the working. A Ferris wheel with 12 equally spaced spokes, \(30^\circ\) apart, is one real example of angles meeting at a single point. This subtopic is part of Year 7 Angles.
Every question with a fully worked solution.
- Angles at a Point and Vertically Opposite Angles - Video - Anlges at a Point and Vertically Opposite Angles Watch
Theory
Angles at a point go all the way round, so they add to \(360^\circ\). When two straight lines cross, the angles opposite each other are vertically opposite angles, and they are equal. This Year 7 maths page shows how to use these facts, with a reason for each step.
A right angle is \(90^\circ\). A straight angle is \(180^\circ\). A complete revolution is \(360^\circ\).
Angles at a point share one vertex and fill a complete revolution. Angles at a point add to \(360^\circ\).
Two straight lines that cross make four angles. The angles opposite each other are vertically opposite angles. Vertically opposite angles are equal.
Where two lines cross, adjacent angles sit on a straight line, so angles on a straight line add to \(180^\circ\).
Reasons to write:
How to find an unknown angle at a point
- Write the reason: angles at a point add to \(360^\circ\).
- Add the known angles (a square counts as \(90^\circ\)).
- Subtract from \(360\) (or solve the equation).
- Check that all the angles add to \(360\).
\(a^\circ\) is opposite \(124^\circ\): vertically opposite angles are equal. \(b^\circ\) is next to \(a^\circ\) on a straight line: angles on a straight line add to \(180^\circ\).
| \(a\) | \(=\) | \(124\) |
| \(b\) | \(=\) | \(180 - 124 = 56\) |
\(a = 124\), \(b = 56\). Check: \(124 + 56 + 124 + 56 = 360\).
The angle and the reflex angle make a revolution: angles at a point add to \(360^\circ\).
| \(360 - 82\) | \(=\) | \(278\) |
Reflex \(\angle ABC = 278^\circ\)
The square shows \(90^\circ\). Angles at a point add to \(360^\circ\): \(90 + 115 + 68 = 273\).
| \(x + 273\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(360 - 273 = 87\) |
\(x = 87\). Check: \(90 + 115 + 68 + 87 = 360\).
| \(105 + 3x\) | \(=\) | \(360\) |
| \(3x\) | \(=\) | \(255\) |
| \(x\) | \(=\) | \(85\) |
\(x = 85\). Check: \(105 + 85 + 85 + 85 = 360\).
\(\angle CEB\) is opposite \(\angle AED\): vertically opposite angles are equal. \(\angle AEC\) is next to \(\angle AED\) on line \(CD\).
| \(\angle CEB\) | \(=\) | \(120\) |
| \(\angle AEC\) | \(=\) | \(180 - 120 = 60\) |
\(\angle CEB = 120^\circ\) and \(\angle AEC = 60^\circ\).
Angles on a straight line add to \(180^\circ\). Then vertically opposite angles are equal.
| \(2x + (x + 30)\) | \(=\) | \(180\) |
| \(3x\) | \(=\) | \(150\) |
| \(x\) | \(=\) | \(50\) |
| \(y\) | \(=\) | \(50 + 30 = 80\) |
\(x = 50\), \(y = 80\). Check: \(2 \times 50 + 80 = 180\).
| \(360 \div 10\) | \(=\) | \(36\) |
Angles at a point add to \(360^\circ\): each angle is \(36^\circ\).
Common pitfalls
Frequently asked questions
What do angles at a point add to?
\(360^\circ\).
What are vertically opposite angles?
The angles opposite each other where two straight lines cross. They are equal.
Are vertically opposite angles equal?
Yes, always.
What do the angles next to each other at a crossing add to?
\(180^\circ\), because they sit on a straight line.
How do you find a reflex angle at a point?
Subtract the other angle from \(360^\circ\). \(360 - 82 = 278\).