Angles in Polygons and Parallel Lines

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

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1. Angles in parallel lines

The three relationships

  • When a straight line crosses a pair of parallel lines, three named relationships appear
  • Corresponding angles are equal. They sit in matching positions at the two crossings
Two parallel lines marked with arrows and cut by a third line, drawn twice: in the first a pair of corresponding angles sitting in matching positions at the two crossings is shaded, and in the second a pair of alternate angles on opposite sides of the crossing line between the parallels
Source: Angles in parallel lines by Save My Exams
  • Alternate angles are equal. They sit on opposite sides of the crossing line, between the parallel lines
  • Co-interior angles add to 180°. They sit on the same side of the crossing line, between the parallel lines
  • The syllabus also calls co-interior angles supplementary, which means they add to 180°
The same pair of parallel lines cut by a third, with two co-interior pairs shaded in turn: each pair sits on the same side of the crossing line between the parallels, and the two shaded angles in each pair add to 180 degrees
Source: Angles in parallel lines by Save My Exams
  • Vertically opposite angles and angles on a straight line still apply, and combining them with the three above solves almost everything
A line crossing two parallel lines with all eight angles filled in: every angle is either x or 180 − x, arranged so the matching ones at each crossing are equal and any two next to each other add to 180
Source: Angles in parallel lines by Save My Exams
  • Arrows on a diagram are what tell you the lines are parallel, so look for them before using any of these
Two parallel lines marked with arrows and crossed by a sloping line, with one angle given as 64 degrees and two others labelled a and b to be found from it
Source: Angles in parallel lines by Save My Exams

Naming the reason

  • Questions ask for the angle and a geometrical reason, with a mark for each
  • Give the full name: "alternate angles are equal", "corresponding angles are equal", "co-interior angles add to 180°"
  • The F, Z and C shapes are useful for spotting the relationship but are not accepted as the reason on their own
  • Abbreviating the name loses the mark, so write it out in full

Worked example

Two angles with reasons

A straight line crosses two parallel lines. One of the angles it makes is 118°. Find the angle corresponding to it, and the angle co-interior with that one, giving a reason for each.

Two parallel horizontal lines, each marked with a single arrowhead to show they are parallel, crossed by a sloping transversal. At the upper crossing, the angle between the line to the right and the transversal going up to the left is marked 118 degrees. No other angle is marked.

Solution:

  • The corresponding angle sits in the matching position at the other crossing: above the lower line, to the right of the crossing line
  • Corresponding angles are equal, so that angle is 118°
  • It lies between the parallel lines, so it has a co-interior partner: the angle at the upper crossing on the same side of the crossing line, also between the parallel lines
  • Co-interior angles add to 180°, so that angle is 180 − 118 = 62°
  • Check: the 62° angle and the given 118° lie on a straight line at the upper crossing, and 118 + 62 = 180

2. Angles in polygons

Interior and exterior angles

  • An interior angle sits inside the polygon at a corner
  • An exterior angle sits between one side and the extension of the next side
  • At every corner the interior and exterior angles lie on a straight line, so they add to 180°
  • Whatever the polygon and however many sides it has, its exterior angles total 360°
  • For the inside angles the total depends on n: a polygon with n sides splits into n − 2 triangles, so they come to 180° × (n − 2)
    • A triangle gives 180°, a quadrilateral 360°, a pentagon 540° and a hexagon 720°
An irregular hexagon with each of its six interior angles labelled a to f in blue, and each side extended with a dashed line so the exterior angle beside it, p to u, is marked in red, the interior and exterior angle at each corner lying on a straight line
Source: Angles in polygons by Save My Exams

Regular polygons

  • A regular polygon has all sides equal and all angles equal, so its angles can be found by dividing
  • Each exterior angle is 360° ÷ n, and this is usually the quicker route
  • Each interior angle is then 180° minus that exterior angle
  • The same result comes from 180(n − 2) ÷ n, so either method is acceptable
  • Working backwards, n = 360° ÷ exterior angle; when a question gives the interior angle, take it from 180° first
Regular polygonSidesInterior sumEach interiorEach exterior
Equilateral triangle3180°60°120°
Square4360°90°90°
Pentagon5540°108°72°
Hexagon6720°120°60°
Octagon81080°135°45°
Decagon101440°144°36°

Worked example

Interior angle from the number of sides

A regular polygon has 40 sides. Find the size of each of its interior angles.

Solution:

  • The exterior angles of any polygon add to 360°, and in a regular polygon they are all equal
  • So each exterior angle is 360 ÷ 40 = 9°
  • The interior and exterior angles at each corner lie on a straight line, so they add to 180°
  • Each interior angle is 180 − 9 = 171°
  • Check with the other method: 180 × (40 − 2) = 6840, and 6840 ÷ 40 = 171

Worked example

Finding the number of sides

Each exterior angle of a regular polygon is 24°. Work out the number of sides, and the size of each interior angle.

Solution:

  • The exterior angles of any polygon add to 360°
  • For a regular polygon they are all equal, so the number of sides is 360 ÷ 24
  • The polygon has 15 sides
  • The interior and exterior angles at each corner lie on a straight line
  • Each interior angle is 180 − 24 = 156°
  • Check with the other method: 180 × (15 − 2) = 2340, and 2340 ÷ 15 = 156

Worked example

Number of sides from an interior angle

Each interior angle of a regular polygon is 168°. Find the number of sides.

Solution:

  • The interior and exterior angles at each corner lie on a straight line, so each exterior angle is 180 − 168 = 12°
  • The exterior angles add to 360° and are all equal, so the number of sides is 360 ÷ 12
  • The polygon has 30 sides
  • Check with the other method: 180(n − 2) ÷ n = 168 gives 180n − 360 = 168n, so 12n = 360 and n = 30

Dividing 360 by the interior angle itself gives 360 ÷ 168, which is not a whole number: a sign that the step to the exterior angle was missed.