Unit 5 · Angles and constructions

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5.1 A sum of 360° Geometry and measure

  • A full turn is 360°, so the angles around a point total 360°.
  • A straight line is half a turn, so the angles on a straight line total 180°.
  • The angles of a triangle total 180°.
  • The angles of a quadrilateral total 360° — draw one diagonal and it splits into two triangles.
  • To find a missing angle: add the ones you know, then subtract from the total.
  • Acute < 90°; right = 90°; obtuse is between 90° and 180°; reflex > 180°.
  • Check your answer is sensible: does it look about the right size in the diagram?

Key words: angle  ·  degree (°)  ·  angles at a point  ·  straight line  ·  quadrilateral  ·  acute angle  ·  right angle  ·  obtuse angle  ·  reflex angle

120° + 95° + x = 360°
x = 360° − 215° = 145°
65° + 40° + y = 180°
y = 180° − 105° = 75°
140° + 75° + x = 360°
215° + x = 360°
x = 145° (angles at a point)
55° + 72° + e = 180°

5.2 Intersecting lines Geometry and measure

  • Where two straight lines cross, four angles are made.
  • Vertically opposite angles are equal — the pairs opposite each other.
  • Angles side by side lie on a straight line, so they add to 180°.
  • Because of that, the four angles are a, 180−a, a, 180−a — and they total 360°.
  • If one angle is 90° then all four are, and the lines are perpendicular.
  • Parallel lines never meet. A line crossing them is a transversal.
  • On parallel lines, corresponding angles (F shape) are equal and alternate angles (Z shape) are equal.
  • Always state the reason: “vertically opposite angles are equal”, or “angles on a straight line add to 180°”.

Key words: intersecting lines  ·  vertically opposite angles  ·  perpendicular  ·  parallel  ·  transversal  ·  corresponding angles  ·  alternate angles

Four angles at a crossing
Two pairs, each pair equal
If one angle is 62°
the angle opposite it is also 62°
a, 180−a, a, 180−a
Total: 2a + 2(180−a) = 360° ✓
90°, 90°, 90°, 90°
The little square means exactly 90°

5.3 Drawing lines and quadrilaterals Geometry and measure

  • Measure and draw angles with a protractor; measure lengths with a ruler.
  • Put the centre of the protractor on the vertex and one arm on a zero, then read that same scale round to the other arm.
  • Check every reading against how the angle looks: an acute angle cannot be 130°.
  • Parallel lines are marked with matching arrows; perpendicular lines with a small square.
  • Square: four equal sides, four right angles. Rectangle: four right angles.
  • Rhombus: four equal sides. Parallelogram: two pairs of parallel sides.
  • Trapezium: exactly one pair of parallel sides. Kite: two pairs of adjacent equal sides.
  • In any parallelogram, adjacent angles add to 180° and opposite angles are equal.

Key words: protractor  ·  parallel  ·  perpendicular  ·  square  ·  rectangle  ·  rhombus  ·  parallelogram  ·  trapezium  ·  kite

An angle that looks acute must read under 90°
If you read 130° for it, you used the wrong scale — it is 50°
To draw 65°: draw the base line, mark 65 on the scale, join up
→→ on two lines means they are parallel
⌐ in a corner means exactly 90°
square 4 equal sides, 4 right angles
rectangle 4 right angles
rhombus 4 equal sides