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Geometry & measures · Angles & polygons

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Angle rules & giving reasons

Why angles at a point make 360°, angles on a straight line make 180°, and vertically opposite angles are equal; the triangle and quadrilateral angle sums; isosceles triangles; and how to write the reasons examiners insist on.

Geometry & measures · Angles & polygons

Angle rules & giving reasons

Why angles at a point make 360°, angles on a straight line make 180°, and vertically opposite angles are equal; the triangle and quadrilateral angle sums; isosceles triangles; and how to write the reasons examiners insist on.

Why it works

An angle measures turn. A full turn is 360°360°, so angles meeting at a point share out the full turn: angles at a point add up to 360°. A straight line is exactly half a turn, so angles on a straight line add up to 180°.

Vertically opposite angles are equal — and you can see why. When two straight lines cross they make four angles. Call two neighbouring ones aa and bb: they sit on a straight line, so a+b=180°a + b = 180°. But bb and the angle opposite aa also sit on a straight line, so that opposite angle is 180°−b=a180° - b = a. The two angles facing each other across the crossing are forced to be equal.

Angles in a triangle add up to 180°. Draw a line through one vertex parallel to the opposite side. The two base angles reappear up at the top as alternate angles, and the three angles at the top sit on a straight line — so the three angles of the triangle total 180°180°. A quadrilateral cuts into two triangles along a diagonal, so its angles add up to 2×180°=360°2 \times 180° = 360°.

Isosceles triangles: equal sides mean equal angles. The two angles that are equal are the ones opposite the equal sides — the "base angles". In a figure, the dash marks on two sides are the signal. If the angle between the equal sides is 40°40°, the other two share 180°−40°=140°180° - 40° = 140° equally: 70°70° each.

Reasons are part of the answer. GCSE questions say *"give a reason for each stage of your working"* — and the reason must name the fact in full:
  • "angles on a straight line add up to 180°"
  • "angles at a point add up to 360°"
  • "vertically opposite angles are equal"
  • "angles in a triangle add up to 180°"
  • "base angles of an isosceles triangle are equal"
A calculation with no reason, or a vague reason like "angle rule", loses the communication marks even when the number is right. One reason per stage, written next to the stage it justifies.