Every angle-chasing problem is built from a handful of core facts, and each one must be stated as a reason to earn full marks:

• Angles at a point sum to $360^\circ$.
• Angles on a straight line sum to $180^\circ$.
• Vertically opposite angles are equal.
• Angles in a triangle sum to $180^\circ$.
• Angles in a quadrilateral sum to $360^\circ$.

When a transversal crosses two parallel lines, three relationships appear:

        ╱              Alternate angles (Z-shape):     equal
  ─────╱───── Line 1    Corresponding angles (F-shape): equal
      ╱                 Co-interior angles (C-shape):   sum to 180°
  ───╱────── Line 2
    ╱

Tracing the letter shape with your finger over the diagram is a reliable way to identify which rule applies under time pressure — if it looks like a "Z", the angles are equal; if it looks like a "C" or "U", they sum to $180^\circ$.

For any polygon with $n$ sides:

The $(n-2)\times180^\circ$ rule comes from splitting the polygon into triangles by drawing diagonals from one vertex: an $n$-sided polygon always splits into exactly $n-2$ triangles this way, and each triangle contributes $180^\circ$ to the total. The exterior-angle sum of $360^\circ$ holds for any polygon, regular or not, because walking all the way around the outside and turning through each exterior angle in turn brings you back to face your starting direction — one full turn, every time.

For a regular polygon, divide by $n$ to get a single angle:

This division by $n$ only works when the polygon is regular — for an irregular polygon, only the angle sum is fixed, not the individual angles, since the sides and angles aren't all equal.

Worked example. A regular polygon has an exterior angle of $24^\circ$. Find (a) its number of sides, (b) its interior angle.

(a) $n = 360^\circ \div 24^\circ = 15$ sides.

(b) Interior and exterior angles lie on a straight line: $180^\circ - 24^\circ = 156^\circ$.

Worked example. In the diagram, two parallel lines are crossed by a transversal, creating a co-interior angle of $x + 30^\circ$ and its partner of $2x$. Find $x$.

Co-interior angles sum to $180^\circ$: