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Year 6 Maths Angles

Vertically opposite angles (intersecting lines)

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Theory

When two straight lines cross, they make four angles. The angles opposite each other (vertically opposite angles) are equal, and angles next to each other sit on a straight line, so they add to \(180^\circ\). This Year 6 maths page shows how to find all four angles from one, and why opposite angles must be equal.

Two straight lines that cross intersect. The point where they cross is the vertex of four angles.

Vertically opposite angles are across from each other at the crossing, like the letters a and c in an X. They are equal.

Angles next to each other at the crossing sit on one of the straight lines, so they add to \(180^\circ\).

Why opposite angles are equal: if \(a + b = 180^\circ\) and \(b + c = 180^\circ\), then \(a\) and \(c\) are both \(180^\circ - b\), so \(a = c\).

All four angles add to \(360^\circ\), because they go all the way round the point.

Two lines crossing with one angle given Two straight lines cross. One angle is 48 degrees; going round, the other angles are labelled a, b and c. Angle b is opposite the 48 degree angle. 48° a b c
\(b\) is opposite \(48^\circ\), so \(b = 48^\circ\). \(a\) and \(c\) are each \(180 - 48 = 132^\circ\).
Pairs of vertically opposite angles Two straight lines cross making angles p, q, r and s in order round the point. The arcs mark p and r, which are opposite each other. p q r s
\(p\) and \(r\) are opposite, so \(p = r\). Likewise \(q = s\).

At a crossing of two straight lines:

\[ \text{opposite angles are equal: } p = r, \; q = s \]
opposite angles are equal p = r , q = s
\[ \text{neighbours: } p + q = 180^\circ \]
neighbours p + q =180
\[ p + q + r + s = 360^\circ \]
p + q + r + s =360
Opposite means across the point. Not the angle next to it: that one is 180° minus the angle.
One angle gives all four. Once you know one angle, the others are the same or 180° minus it.

Finding all four angles from one

  1. Opposite angle: the same size.
  2. Neighbouring angles: \(180^\circ\) minus the known angle.
  3. Check: the four angles add to \(360^\circ\).
Example 1 — Opposite angle
Use the first diagram. Find angle \(b\).
Solution

\(b\) is vertically opposite the \(48^\circ\) angle, so \(b = 48^\circ\).

Example 2 — Neighbouring angles
Use the first diagram. Find angles \(a\) and \(c\).
Solution
\(180 - 48\)\(=\)\(132\)
180−48=132

\(a = c = 132^\circ\).

Example 3 — Check the total
Check the four angles of the first diagram.
Solution
\(48 + 132 + 48 + 132\)\(=\)\(360\)
48+132+48+132=360

They make \(360^\circ\), a full turn.

Example 4 — All four from one
Two lines cross and one angle is \(115^\circ\). Find the other three.
Solution
\(180 - 115\)\(=\)\(65\)
180−115=65

The opposite angle is \(115^\circ\); the two neighbours are \(65^\circ\) each.

Example 5 — Crossing roads
Two straight roads cross. One angle between them is \(58^\circ\). Find the other three angles.
Solution
\(180 - 58\)\(=\)\(122\)
180−58=122

\(58^\circ\), \(122^\circ\) and \(122^\circ\).

Example 6 — One angle bigger
Two lines cross. One angle is \(30^\circ\) more than the angle next to it. Find all four angles.
Solution
\((180 - 30) \div 2\)\(=\)\(75\)
\(75 + 30\)\(=\)\(105\)
(180−30)÷2=75,75+30=105

Angles next to each other add to \(180^\circ\), and opposite angles are equal: \(75^\circ\), \(105^\circ\), \(75^\circ\), \(105^\circ\).

Example 7 — Three times as big
Two lines cross. One angle is \(3\) times its neighbour. Find all four angles.
Solution
\(180 \div 4\)\(=\)\(45\)
\(3 \times 45\)\(=\)\(135\)
45,135,45,135

\(45^\circ\), \(135^\circ\), \(45^\circ\), \(135^\circ\).

Example 8 — Explain
Use the second diagram. Explain why \(p = r\).
Solution

\(p + q = 180^\circ\) (straight line) and \(q + r = 180^\circ\) (straight line). So \(p\) and \(r\) are both \(180^\circ - q\), which means \(p = r\).

Example 9 — Find the mistake
Two lines cross with one angle of \(37^\circ\). Ben says all four angles are \(37^\circ\). What is right?
Solution
\(180 - 37\)\(=\)\(143\)
180−37=143

Only the opposite angle is \(37^\circ\). The two neighbours are \(143^\circ\).

Common pitfalls

Making every angle equal. Only the opposite angle is equal; neighbours add to 180°.
Choosing the neighbour as opposite. Opposite angles are across the point, not next to each other.
Lines that bend. The facts only work when both lines are straight.
Forgetting to check. All four angles must make 360°.

Frequently asked questions

What are vertically opposite angles?

The angles across from each other where two straight lines cross.

Are vertically opposite angles equal?

Yes, always.

What do neighbouring angles at a crossing add to?

\(180^\circ\), because they sit on a straight line.

If one angle is 115°, what are the others?

\(115^\circ\), \(65^\circ\) and \(65^\circ\).

Why are opposite angles equal?

Each one is 180° minus the same neighbouring angle.

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