Angles in Polygons & Parallel Lines Notes

When two straight lines cross they form two pairs of vertically opposite angles, which are equal. Adjacent angles along a line add to $180°$.

A line crossing a pair of parallel lines (a transversal) creates:

• Corresponding angles: equal, in the same position at each crossing (F shape)
• Alternate angles: equal, on opposite sides of the transversal between the lines (Z shape)
• Co-interior angles: add to $180°$, on the same side between the lines (C shape)

These facts need the lines to be marked or stated as parallel.

For any polygon with $n$ sides, drawing diagonals from one vertex gives $n-2$ triangles, so

$\text{sum of interior angles}=(n-2)\times180°$

For a regular polygon, divide by $n$ to find one interior angle: $\dfrac{(n-2)\times180°}{n}$. A regular hexagon has sum $720°$ and each angle $120°$.

The exterior angle at a vertex is the angle between a side and the extension of the next side. Interior $+$ exterior $=180°$.

For a regular polygon, exterior angle $=\dfrac{360°}{n}$. Often the fastest route is to find the exterior angle first, or use $n=\dfrac{360°}{\text{exterior angle}}$ to find the number of sides.

Exam questions chain facts, for example parallel-line angles to find an angle in a polygon, then the angle sum. Mark schemes award marks for correct reasoning, not just the final number.