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Lesson · Practice · Steps

Angles

Read the diagram. Pick the fact. Do the maths. One small step at a time.

The six facts you need

Every angle question uses one of these. You don't need to remember anything else.

1 · Straight lineAngles on a straight line add to 180°
2 · PointAngles at a point add to 360°
3 · Vertically oppositeWhen two lines cross, the angles opposite each other are equal
4 · TriangleThe three angles in a triangle add to 180°
5 · QuadrilateralThe four angles in a four-sided shape add to 360°
6 · Parallel linesF-shape (corresponding) = equal · Z-shape (alternate) = equal · C-shape (co-interior) = add to 180°

Worked example 1 — angles on a straight line

Three angles sit on a straight line: 56°, x, and 72°. Find x.

72°x56°
  1. Read it. Three angles meet on one straight line. We need to find the middle one, x.
  2. The fact. Angles on a straight line add to 180°.
  3. The numbers. We know 56° and 72°. We do not know x.
  4. Add the known angles. 56 + 72 = 128.
  5. Take that from 180. 180 − 128 = 52.
Answer: x = 52°
Watch outSome pupils add all three: 56 + x + 72 = 360. That is wrong. A point is 360°. A straight line is 180°.

Worked example 2 — angles at a point

Four angles meet at a point: 120°, 95°, 78°, and x. Find x.

120°95°78°x
  1. Read it. Four rays leave the same dot. The whole way round is one full turn.
  2. The fact. Angles at a point add to 360°.
  3. The numbers. We know 120°, 95° and 78°. We do not know x.
  4. Add the known angles. 120 + 95 + 78 = 293.
  5. Take that from 360. 360 − 293 = 67.
Answer: x = 67°
Watch outA point goes all the way round (360°). A straight line only goes half-way round (180°). Use 360°, not 180°.

Worked example 3 — vertically opposite angles

Two straight lines cross. One angle is 64°. Find the angle directly opposite, x.

64°x
  1. Read it. Two lines cross at one dot. They make four angles. We need the one opposite the 64°.
  2. The fact. When two lines cross, opposite angles are equal.
  3. Look at the matching ticks. The little tick lines show which two angles are a matching pair.
  4. So they are the same number. x is the same as 64°.
Answer: x = 64°
Watch outPupils sometimes pick the angle next to 64° instead of the one opposite. Next-to angles are NOT equal — they add to 180°. The opposite one (with the matching tick) is the equal one.

Worked example 4 — angles in a triangle

A triangle has two angles of 47° and 68°. Find the third angle, x.

47°68°x
  1. Read it. A triangle. Two angles known. One missing.
  2. The fact. The three angles in a triangle add to 180°.
  3. The numbers. We know 47° and 68°. We do not know x.
  4. Add the known angles. 47 + 68 = 115.
  5. Take that from 180. 180 − 115 = 65.
Answer: x = 65°
Watch outTriangles add to 180°. Squares and rectangles add to 360°. Do not mix them up.

Worked example 5 — angles in a quadrilateral

A four-sided shape has angles 85°, 95°, 110° and x. Find x.

85°95°110°x
  1. Read it. A shape with four corners. Three angles known. One missing.
  2. The fact. The four angles in a four-sided shape add to 360°.
  3. The numbers. We know 85°, 95° and 110°. We do not know x.
  4. Add the known angles. 85 + 95 + 110 = 290.
  5. Take that from 360. 360 − 290 = 70.
Answer: x = 70°
Watch outIt is easy to forget the total is 360° for a four-sided shape, not 180°. A triangle has 3 sides → 180°. A four-sided shape has 4 sides → 360°.

Worked example 6 — parallel lines — corresponding (F-shape)

Two parallel lines are crossed by another line. One angle is 58°. Find x, the matching (corresponding) angle.

58°x
  1. Read it. Two parallel lines. A line crosses both of them.
  2. Spot the shape. The 58° and x make an F-shape (look for the F lying down). That means they are corresponding angles.
  3. The fact. Corresponding angles are equal.
  4. The answer is the same number. x is the same as 58°.
Answer: x = 58°
Watch outDifferent shapes give different rules: F = equal · Z = equal · C = add to 180°. Always check the shape first.

Worked example 7 — parallel lines — alternate (Z-shape)

Two parallel lines are crossed by another line. One angle is 64°. Find the marked angle, y.

64°y
  1. Read it. Two parallel lines. A line crosses both of them.
  2. Spot the shape. The 64° and y make a Z-shape. That means they are alternate angles.
  3. The fact. Alternate angles are equal.
  4. The answer is the same number. y is the same as 64°.
Answer: y = 64°
Watch outThe Z-shape needs both angles to be INSIDE the parallel lines, one on each side of the crossing line. If they are not inside, it is not a Z.

Worked example 8 — parallel lines — co-interior (C-shape)

Two parallel lines are crossed by another line. One angle is 72°. Find x, the co-interior angle (on the same side, both inside the parallel lines).

72°x
  1. Read it. Two parallel lines. A line crosses both of them. Two angles sit on the same side of the crossing line, both between the parallels.
  2. Spot the shape. The 72° and x make a C-shape. That means they are co-interior angles.
  3. The fact. Co-interior angles add to 180° (they are NOT equal).
  4. Take the known angle from 180. 180 − 72 = 108.
Answer: x = 108°
Watch outF and Z give EQUAL angles. C is the odd one out — co-interior angles ADD to 180°. The C-shape is the "add up" one.
Practice · Section A
Lines, points and crossings
Straight line · Point · Vertically opposite
Practice · Section B
Triangles and quadrilaterals
Triangle adds to 180° · Quadrilateral adds to 360°
Practice · Section C
Parallel lines
F-shape · Z-shape · C-shape

Ready for more? 100 questions

When you have done these 12, you are ready for the bigger practice paper.

It has 100 questions. They start very easy and get harder near the end. Every question is still Grade 3 — nothing harder. You can stop at any time and come back.

Open the 100-question paper →

It opens in this same place. The lesson stays open behind it.

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