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Cambridge IGCSE0580

Angles

Mathematics 0580 Chapter Notes

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Angles
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In this chapter

  1. 1Angles at a point, on a line and vertically opposite
  2. 2Triangles, quadrilaterals and three-letter notation
  3. 3Angles in parallel lines
  4. 4Regular polygons
  5. 5Number of sides of a regular polygon
  6. 6Irregular polygons

1. Angles at a point, on a line and vertically opposite

Angles at a point sum to 360°. Angles on a straight line add to 180°.

When two straight lines cross, the angles opposite each other are equal: vertically opposite angles are equal. Give the reason in full. Write the property, not just the sum.

at a point: a + b + c = 360°

on a straight line: a + b = 180°

Examiner insight

“The incorrect reasons often included a comment about there being 360° in a circle or angles sum to 360° instead of, angles at a point sum to 360°.” “Of the two reasons, many gained the mark for angles on a straight line added to 180°, although omission of one of the key words, ‘angles’, ‘line’ and ‘180’ lost that mark.”

Worked example 13 marks

Two straight lines cross. One of the four angles is 112°. Work out the other three angles. Give a geometrical reason for each.

  1. 1

    the angle opposite 112° is 112°: vertically opposite angles are equal [1]

  2. 2

    the angle next to 112° is 180 - 112 = 68°: angles on a straight line add to 180° [1]

  3. 3

    the last angle is 68°: vertically opposite angles are equal [1]

2. Triangles, quadrilaterals and three-letter notation

The angle sum of a triangle is 180° and the angle sum of a quadrilateral is 360°.

Angle ABC means the angle at B, between the lines BA and BC. The middle letter is the corner.

In an isosceles triangle, two angles are equal (the angles opposite the equal sides). In the reason, say that the angles are equal, not the sides.

triangle: 180° quadrilateral: 360°

Examiner insight

“Again, the correct geometrical reason was less successfully stated, and candidates should note that the full reason should be stated and not partial reasons such as ‘a quadrilateral adds to 180’.” “The final mark was often lost by referring to sides being equal, already shown in the diagram, rather than angles, which was the required first step to find angle y.”

Worked example 13 marks

Do not use a calculator. Triangle ABC has AB = AC and angle BAC = 40°. Work out angle ABC. Give a reason for each step.

  1. 1

    angle ABC + angle ACB = 180 - 40 = 140°: angles in a triangle add to 180° [1]

  2. 2

    angle ABC = angle ACB: the base angles of an isosceles triangle are equal [1]

  3. 3

    angle ABC = 140 ÷ 2 = 70° [1]

3. Angles in parallel lines

A line crossing two parallel lines makes pairs of equal or supplementary angles.

Corresponding angles are equal. Alternate angles are equal. Co-interior angles sum to 180°.

Name the property in the reason. A letter shape such as Z only helps you find the pair.

Examiner insight

“Z angles is not accepted as this is the shape that is made with the parallel lines and not the proper reason – alternate angles.” “Acceptable vocabulary in reasoning for b was less common although those gaining credit usually did so by referring to the ‘interior angles’, or better still that ‘co-interior angles sum to 180°’.”

Worked example 14 marks

Do not use a calculator. AB and CD are parallel lines. P lies on AB and Q lies on CD. A and C are on the same side of the line PQ. Angle APQ = 118°.

(a) Find angle PQC and give a reason.

(b) Find angle PQD and give a reason.

  1. 1

    (a) angle PQC = 180 - 118 [1]

  2. 2

    = 62°: co-interior angles sum to 180° [1]

  3. 3

    (b) angle PQD = angle APQ = 118° [1]

  4. 4

    alternate angles are equal [1]

4. Regular polygons

In a regular polygon all the interior angles are equal and all the exterior angles are equal. The sum of the exterior angles is 360°, so each exterior angle of a regular polygon with n sides is 360 ÷ n. An interior angle and its exterior angle add to 180°, not 360°.

The angle sum of a polygon with n sides is (n - 2) × 180°. Dividing it by n also gives one interior angle.

exterior angle = (360°)/n

interior angle = 180° - exterior angle

Examiner insight

“By far the most common error was to think that the interior and exterior angles summed to 360 instead of 180.”

Worked example 12 marks

Work out the size of an interior angle of a regular 10-sided polygon.

  1. 1

    exterior angle = 360 ÷ 10 = 36° [1]

  2. 2

    interior angle = 180 - 36 = 144° [1]

5. Number of sides of a regular polygon

To find the number of sides from an interior angle, find the exterior angle first: subtract the interior angle from 180°.

Then divide 360° by the exterior angle.

Do not put the interior angle equal to (n - 2) × 180. That expression is the sum of all the interior angles, not one of them.

n = (360°)/(exterior angle)

Examiner insight

“Those who found the exterior angle first tended to be more successful. Those who attempted the sum of the interior angles were less successful and tended to start with the incorrect formula of (n – 2) × 180 = 171.”

Worked example 12 marks

Find the number of sides of a regular polygon with interior angle 140°.

  1. 1

    exterior angle = 180 - 140 = 40° [1]

  2. 2

    n = 360 ÷ 40 = 9 [1]

6. Irregular polygons

The angles of an irregular polygon are not all equal. Do not assume a polygon is regular unless the question says so.

The angle sum (n - 2) × 180° still holds. For a hexagon it is 4 × 180 = 720°, not 6 × 180.

To find a missing angle, add the angles you know and subtract from the angle sum.

missing angle = (n - 2) × 180° - sum of the other angles

Examiner insight

“Many candidates multiplied 5 by 115 correctly but then had no strategy for finding the sum of angles of a hexagon, with many thinking it was 6 × 180 = 1080.” “A significant number of candidates simply assumed that the polygon was regular and gave 115 as the answer.”

Worked example 13 marks

Do not use a calculator. Five of the angles of a hexagon are each 115°. Work out the sixth angle.

  1. 1

    angle sum of a hexagon = (6 - 2) × 180 = 720° [1]

  2. 2

    five angles: 5 × 115 = 575° [1]

  3. 3

    sixth angle = 720 - 575 = 145° [1]

Recap

  • Angles at a point sum to 360°. Angles on a straight line add to 180°.
  • Vertically opposite angles are equal.
  • Angle sum of a triangle is 180°. Angle sum of a quadrilateral is 360°.
  • Angle ABC is the angle at B. In an isosceles triangle two angles are equal.
  • Parallel lines: corresponding equal, alternate equal, co-interior sum to 180°.
  • Regular polygon: exterior angle = 360 ÷ n. Interior angle = 180° - exterior angle.
  • Number of sides = 360° ÷ exterior angle. Angle sum = (n - 2) × 180°.
  • Extended: the angle sum also holds for an irregular polygon, but its angles are not equal.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Three angles meet at a point. Two of them are 130° and 95°. Work out the third angle and give a reason.2 marks
  2. Two straight lines cross. One of the four angles is 38°. Work out the size of an angle next to it and give a reason.2 marks
  3. Do not use a calculator. Three angles of a quadrilateral are 85°, 110° and 72°. Work out the fourth angle.2 marks
  4. Triangle PQR has PQ = PR and angle PQR = 52°. Work out angle QPR.2 marks
  5. PQ and RS are parallel lines. A line crosses PQ at X and RS at Y. P and S are on opposite sides of XY. Angle PXY = 73°. Find angle XYS and give a reason.2 marks
  6. Do not use a calculator. PQ and RS are parallel lines. A line crosses PQ at X and RS at Y. P and R are on the same side of XY. Angle PXY = 73°. Find angle XYR and give a reason.2 marks
  7. Do not use a calculator. A road sign is a regular octagon. Its vertices, in order, are A, B, C, D, E, F, G and H. Work out the size of angle BAC.4 marks
  8. A pentagon has four angles of 100°, 120°, 105° and 115°. Work out the fifth angle.3 marks
  9. The floor of a bandstand is a regular polygon. Each interior angle is 8 times as large as each exterior angle. (a) Work out the number of sides of the polygon. (b) Work out the sum of its interior angles.4 marks
  10. Do not use a calculator. A floor is covered with three kinds of tile: squares, regular hexagons and a third regular polygon. At every point where tiles meet, one square, one regular hexagon and one tile of the third kind fit together with no gaps. Work out the number of sides of the third kind of tile.5 marks

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