Year 7 Maths Angles

Angles at a Point and Vertically Opposite Angles

Q: What do the angles next to each other at a crossing add to?

180°, because they sit on a straight line.

Q: How do you find a reflex angle at a point?

Subtract the other angle from 360°. 360 - 82 = 278.

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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 \(180^\circ\).

The three angles fill one full turn: \(110 + 120 + 130 = 360\).

Matching colours show the vertically opposite pairs: \(35 = 35\) and \(145 = 145\). Each red angle and the green angle next to it add to \(180^\circ\).

Reasons to write:

\[ \text{Angles at a point add to } 360^\circ \]

Angles at a point add to 360

\[ \text{Vertically opposite angles are equal} \]

Vertically opposite angles are equal

\[ \text{Angles on a straight line add to } 180^\circ \]

Angles on a straight line add to 180

          Method. Write the reason, then the equation. Add the known angles and subtract from \(360\): \(95 + 140 + x = 360\), so \(x = 360 - 235 = 125\).
        

          Equal angles. Angles marked with the same pronumeral are equal: \(x + x = 2x\). Divide to find \(x\).
        

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 \(360\).

Example 1 — Two lines cross

Two straight lines cross. Find the values of \(a\) and \(b\).

Solution

\(a^\circ\) 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 = 180 - 124 = 56

\(a = 124\), \(b = 56\). Check: \(124 + 56 + 124 + 56 = 360\).

Example 2 — A reflex angle

Find the size of reflex \(\angle ABC\), where \(\angle ABC = 82^\circ\).

Solution

The angle and the reflex angle make a revolution: angles at a point add to \(360^\circ\).

\(360 - 82\)

\(=\)

\(278\)

360 - 82 = 278

Reflex \(\angle ABC = 278^\circ\)

Example 3 — An unknown angle at a point

Find the value of \(x\).

Solution

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 + 273 = 360, x = 87

\(x = 87\). Check: \(90 + 115 + 68 + 87 = 360\).

Example 4 — Equal angles at a point

One angle at a point is \(105^\circ\). The other three angles are all \(x^\circ\). Find \(x\).

Solution

\(105 + 3x\)	\(=\)	\(360\)
\(3x\)	\(=\)	\(255\)
\(x\)	\(=\)	\(85\)

105 + 3 x = 360, 3 x = 255, x = 85

\(x = 85\). Check: \(105 + 85 + 85 + 85 = 360\).

Example 5 — Opposite angles with names

Lines \(AB\) and \(CD\) cross at \(E\), and \(\angle AED = 120^\circ\). Find \(\angle CEB\) and \(\angle AEC\).

Solution

\(\angle CEB\) is opposite \(\angle AED\): vertically opposite angles are equal. \(\angle AEC\) is next to \(\angle AED\) on line \(CD\).

\(\angle CEB\)	\(=\)	\(120\)
\(\angle AEC\)	\(=\)	\(180 - 120 = 60\)

CEB = 120, AEC = 180 - 120 = 60

\(\angle CEB = 120^\circ\) and \(\angle AEC = 60^\circ\).

Example 6 — Pronumerals on crossing lines

Two straight lines cross. Adjacent angles are \((x + 30)^\circ\) and \(2x^\circ\), and \(y^\circ\) is opposite \((x + 30)^\circ\). Find \(x\) and \(y\).

Solution

Angles on a straight line add to \(180^\circ\). Then vertically opposite angles are equal.

\(2x + (x + 30)\)	\(=\)	\(180\)
\(3x\)	\(=\)	\(150\)
\(x\)	\(=\)	\(50\)
\(y\)	\(=\)	\(50 + 30 = 80\)

2 x + x + 30 = 180, 3 x = 150, x = 50, y = 80

\(x = 50\), \(y = 80\). Check: \(2 \times 50 + 80 = 180\).

Example 7 — Spokes of a wheel

A wheel has \(10\) equally spaced spokes meeting at the centre. What is the angle between two spokes next to each other?

Solution

\(360 \div 10\)

\(=\)

\(36\)

360 \div 10 = 36

Angles at a point add to \(360^\circ\): each angle is \(36^\circ\).

Common pitfalls

Opposite, not next door. The angle next to \(35^\circ\) is \(145^\circ\): they are adjacent and add to \(180^\circ\). They are not equal.

Using 180 instead of 360. Angles at a point make a full turn. Only angles on a straight line add to \(180^\circ\).

Forgetting the right angle. A small square means \(90^\circ\), even with no number written. Put it in the sum.

Counting x once. Two angles marked \(x^\circ\) give \(x + x = 2x\), not \(x\).

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 \(360^\circ\). \(360 - 82 = 278\).

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Complementary and Supplementary Angles

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Parallel and Perpendicular Lines

Angles at a Point and Vertically Opposite Angles

📖 Theory

How to find an unknown angle at a point

Common pitfalls

Frequently asked questions

Theory