0580

Angles in Polygons and Parallel Lines

Geometry

Four facts that start every question

  • Angles meeting at a point add to 360°
  • Angles on one side of a straight line add to 180°
  • Vertically opposite angles, formed where two straight lines cross, are equal
Two straight lines crossing, drawn twice: in the first the pair of opposite angles above and below the crossing point is shaded, and in the second the other opposite pair, left and right, is shaded, so each pair is equal
Source: Basic angle properties by Save My Exams
  • The angles inside a triangle add to 180°, and the angles inside a quadrilateral add to 360°
  • Naming uses three letters with the vertex in the middle, so angle ABC is the angle at B, and it means the acute or obtuse angle rather than the reflex one
  • Diagrams are usually marked "NOT TO SCALE", so work from the numbers and the markings rather than from how the picture looks
Three straight lines crossing at one point, marked not to scale, with angles of 25 degrees and 98 degrees given and two more, x and y, to be found from the facts that angles at a point add to 360 and angles on a line add to 180
Source: Basic angle properties by Save My Exams

Filling in what you can

  • Mark every angle you can work out on the diagram, even ones the question has not asked for
  • One of them is almost always the step that unlocks the angle you actually want
  • A question worth more than one mark expects the intermediate angles to appear in the working
A triangle standing on a straight line with 60 degrees and 130 degrees marked outside it and x inside at the apex, and the two angles y and z between the triangle and the line filled in as the intermediate step
Source: Basic angle properties by Save My Exams
Worked example

Crossing lines and a straight line

Three straight lines meet at a point. One angle is 112°, and x is vertically opposite it. On the straight line through that point sit x, y and an angle of 41°. Find x and y.

Three straight lines crossing at a single point. Going anticlockwise from the right-hand horizontal ray, the first angle is marked 41 degrees, the next is labelled y, and the third, up to the left-hand horizontal ray, is labelled x. Below the horizontal line, the angle vertically opposite x is marked 112 degrees.

Solution:

  • x is vertically opposite the 112° angle, and vertically opposite angles are equal
  • So x = 112
  • The three angles x, y and 41° lie on a straight line, so they add to 180°
  • 112 + y + 41 = 180
  • 153 + y = 180
  • So y = 27
  • Check: 112 + 27 + 41 = 180