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Year 6 Maths Angles

Angles on a straight line

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Theory

A straight line makes a straight angle of \(180^\circ\), half a full turn. So angles that sit side by side on a straight line, sharing a vertex, add to \(180^\circ\). This Year 6 maths page shows how to use this fact to find unknown angles, including with three or more angles, right angles and equal angles, and how to spot a set of angles that cannot lie on a line.

A straight angle is \(180^\circ\). It is half of a full turn (\(360^\circ\)) and two right angles (\(2 \times 90^\circ\)).

Angles on a straight line are angles that sit next to each other on one side of a straight line and share a vertex. They add to \(180^\circ\).

To find an unknown angle, subtract the known angles from \(180^\circ\).

A small square at the vertex marks a right angle (\(90^\circ\)).

If the angles you are given do not add to \(180^\circ\), they cannot all sit on one straight line.

Two angles on a straight line A straight line with a ray from a point on it. The angle on the right is 125 degrees and the angle on the left is marked with a question mark. 125° ?
\(125^\circ + {?} = 180^\circ\), so the missing angle is \(55^\circ\).
Three angles on a straight line A straight line with two rays from one point, making three angles: 55 degrees, 70 degrees and an unknown angle. 55° 70° ?
Three angles: \(180 - 55 - 70 = 55^\circ\).
A right angle on a straight line A straight line with two rays from one point. The middle angle is a right angle marked with a square, the left angle is 28 degrees and the right angle is unknown. ? 28°
With a right angle: \(180 - 90 - 28 = 62^\circ\).

The straight-line fact:

\[ a + b = 180^\circ \]
a + b =180
\[ a + b + c = 180^\circ \]
a + b + c =180
\[ \text{unknown} = 180^\circ - \text{known angles} \]
unknown =180− known angles
180, not 360. Angles on a straight line make half a turn: use \(180^\circ\).
Check the answer. Add all the angles: they must make 180°.

Finding an unknown angle on a straight line

  1. Check that the angles sit together on one straight line at one point.
  2. Add the known angles (a right-angle mark counts as \(90^\circ\)).
  3. Subtract the total from \(180^\circ\).
  4. Check: all the angles add to \(180^\circ\).
Example 1 — Two angles
Use the first diagram. Find the missing angle.
Solution
\(180 - 125\)\(=\)\(55\)
180−125=55

\(55^\circ\).

Example 2 — Three to find
Each angle sits on a straight line next to one unknown angle. Find the unknown angle.
(i) \(84^\circ\)
(ii) \(17^\circ\)
(iii) \(148^\circ\)
Solution
\(180 - 84\)\(=\)\(96\)
\(180 - 17\)\(=\)\(163\)
\(180 - 148\)\(=\)\(32\)
96,163,32

(i) \(96^\circ\)
(ii) \(163^\circ\)
(iii) \(32^\circ\)

Example 3 — Three angles on a line
Use the second diagram. Find the missing angle.
Solution
\(55 + 70\)\(=\)\(125\)
\(180 - 125\)\(=\)\(55\)
180−125=55

\(55^\circ\).

Example 4 — With a right angle
Use the third diagram. Find the missing angle.
Solution
\(90 + 28\)\(=\)\(118\)
\(180 - 118\)\(=\)\(62\)
180−118=62

\(62^\circ\).

Example 5 — Equal angles
Three equal angles sit together on a straight line. How big is each one?
Solution
\(180 \div 3\)\(=\)\(60\)
180÷3=60

\(60^\circ\) each.

Example 6 — Three times as big
Two angles on a straight line: the larger is \(3\) times the smaller. Find both.
Solution

Together they make \(4\) equal parts of \(180^\circ\).

\(180 \div 4\)\(=\)\(45\)
\(3 \times 45\)\(=\)\(135\)
45 and 135

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

Example 7 — A ramp
A ramp meets flat ground at \(24^\circ\). What angle does it make with the ground on the other side?
Solution
\(180 - 24\)\(=\)\(156\)
180−24=156

\(156^\circ\).

Example 8 — Possible or not?
Can angles of \(62^\circ\), \(48^\circ\) and \(75^\circ\) sit together on a straight line?
Solution
\(62 + 48 + 75\)\(=\)\(185\)
62+48+75=185

No. They add to \(185^\circ\), which is \(5^\circ\) too many.

Example 9 — Find the mistake
An angle of \(140^\circ\) sits on a straight line next to an unknown angle. Zoe wrote \(360 - 140 = 220^\circ\). What is right?
Solution
\(180 - 140\)\(=\)\(40\)
180−140=40

Zoe used a full turn. A straight line is \(180^\circ\), so the angle is \(40^\circ\).

Common pitfalls

Using 360°. That is a full turn. A straight line is \(180^\circ\).
Forgetting the right angle. A square mark means \(90^\circ\); include it in the total.
Angles not on the same line. The angles must share a vertex and sit on one side of the same straight line.
Not checking. Add all the angles; they should make exactly 180°.

Frequently asked questions

What do angles on a straight line add to?

\(180^\circ\).

How do I find the missing angle on a straight line?

Subtract the known angles from \(180^\circ\): \(180 - 125 = 55^\circ\).

What if there are three angles?

Add the two you know, then subtract from 180°.

Why is a straight angle 180°?

It is half of a full turn, and a full turn is 360°.

If four equal angles are on a straight line, how big is each?

\(45^\circ\).

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