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Cambridge IGCSE0607

Angle measurement in degrees

International Mathematics 0607 Chapter Notes

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Angle measurement in degrees
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Angle measurement in degrees notes

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In this chapter

  1. 1Measuring and drawing lines and angles
  2. 2Three-figure bearings
  3. 3The bearing of A from B
  4. 4Bearings from an angle to north or south

1. Measuring and drawing lines and angles

Measure and draw lines and angles. A ruler must be used for all straight edges, so a freehand line loses the mark.

An angle named ABC is the angle at the vertex at the middle letter, B. Draw an angle at the point the question names, not at the other end of the line or its midpoint.

A protractor has two scales. Check the reading against the angle you can see: an acute angle is less than 90° and an obtuse angle is more than 90°.

Examiner insight

“Measuring this angle caused some problems as many candidates either read the wrong scale on their protractors (giving answers around 48°) or were not accurate enough with their measuring.” “For questions like these, freehand lines are not acceptable.”

Worked example 12 marks

(a) Draw a line of length 6 cm.

(b) Draw an angle of 70° at A.

  1. 1

    (a) ruled line, correct line length 6 cm [1]

  2. 2

    (b) ruled line from A, the marked end of the line given, correct angle of 70° [1]

2. Three-figure bearings

Bearings are measured clockwise from north and written with three figures, from 000° to 360°. Write 090°, not 90°.

To find the bearing of one point from another, stand at the point named after "from". Face north, then turn clockwise until you face the other point.

Due north is 000°, due east is 090°, due south is 180° and due west is 270°. A compass point such as "west" is not a bearing.

Examiner insight

“Many candidates gave 90 instead of a three-figure bearing.” “Many candidates offered a compass point, usually west or east, rather than a bearing.”

Worked example 11 mark

Q is due west of P. Write down the bearing of Q from P.

  1. 1

    stand at P, face north, turn clockwise to face Q: three quarters of a full turn

  2. 2

    270° [1]

3. The bearing of A from B

If the bearing of B from A is known, the bearing of A from B points the opposite way. The two bearings differ by 180°.

If the first bearing is less than 180°, add 180°. If it is more than 180°, subtract 180°. Sketch it: draw a north line at the point you stand at, and mark the angle you are given.

Examiner insight

“A frequent wrong method was 360 – 45 = 315.” “Candidates are strongly recommended to draw a sketch for any bearings question, marking on the information given in the question.”

Worked example 12 marks

The bearing of B from A is 055°. Find the bearing of A from B.

  1. 1

    north lines are parallel, so the bearing of A from B is 055 + 180 [1]

  2. 2

    235° [1]

4. Bearings from an angle to north or south

A diagram may give an angle from the north line or the south line. Show the angle on a sketch first.

From the north line, on the east side, the bearing is the angle itself. On the west side, subtract the angle from 360°.

From the south line, on the west side, add the angle to 180°. On the east side, subtract it from 180°.

Examiner insight

“Common wrong methods seen included 90 + 35 = 125 or 180 – 35 = 145.” “The first step was to draw a north arrow at Q and to work out which bearing was being asked for.”

Worked example 12 marks

XY makes an angle of 40° with the south line at X, on its west side. Find the bearing of Y from X.

  1. 1

    sketch a north-south line at X and put Y to the west of the south line [1]

  2. 2

    bearing = 180 + 40 = 220° [1]

Recap

  • Use a ruler for every straight edge.
  • Bearings are measured clockwise from north and written with three figures.
  • Due east is 090°, due south 180°, due west 270°.
  • To reverse a bearing, add 180° if it is under 180°, subtract 180° if it is over.
  • Sketch a north line at the point you stand at.

End-of-chapter exercise

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

  1. State the bearing of a point that is due east of P.1 mark
  2. A student measures an acute angle with a protractor and reads 48° on one scale and 132° on the other. State the size of the angle.2 marks
  3. A boat sails due west from a harbour. Write down (a) the bearing of the boat from the harbour (b) the bearing of the harbour from the boat.2 marks
  4. Describe how to draw an angle of 65° at point A on the end of a line AB.3 marks
  5. Do not use a calculator. The bearing of B from A is 040°. Find the bearing of A from B.2 marks
  6. Do not use a calculator. The bearing of Y from X is 250°. Find the bearing of X from Y.2 marks
  7. Do not use a calculator. (a) The line from X to Y makes an angle of 25° with the south line at X, on the east side of it. Find the bearing of Y from X. (b) The line from X to Z makes an angle of 35° with the north line at X, on the west side of it. Find the bearing of Z from X.2 marks
  8. Do not use a calculator. A ship sails from A to B on a bearing of 155°. It then sails due east from B to C. Work out angle ABC.4 marks
  9. Do not use a calculator. A lighthouse L is due north of a harbour H. A yacht Y is on a bearing of 128° from L and on a bearing of 063° from H. (a) Find the bearing of L from Y. (b) Find the bearing of H from Y. (c) Work out angle LYH.5 marks
  10. Do not use a calculator. A robot starts at P facing due north. It turns clockwise through 115° and moves in a straight line to Q. At Q it turns anticlockwise through 160° and moves in a straight line to R. (a) Write down the bearing of Q from P. (b) Find the bearing of R from Q. (c) Find the bearing of Q from R. (d) Work out angle PQR.6 marks

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