Unit 14: Angle Chasing

Mark what you know

Angle chasing has no clever trick. The method is:

1. Write in every angle you already know.
2. Look for one more you can now work out.
3. Write it in. Repeat.

The tools are always the same four: a straight line makes 180°, a point makes 360°, a triangle makes 180°, and parallel lines copy angles across.

The exterior angle shortcut

Did you notice? The exterior angle came out as 40 + 75 — exactly the two angles it is not touching, added together.

That is always true, and it saves a step every time. An exterior angle of a triangle equals the two opposite interior angles.

The angle at the centre is double

Take two points on a circle. Join them to the centre, and join them to any other point on the edge.

The angle at the centre is always twice the angle at the edge.

And because that is true wherever you put the edge point, every angle drawn on the same arc is equal. That is the workhorse of every circle problem.

The angle in a semicircle

Now put those two points at the ends of a diameter. The angle at the centre is a straight line — 180°.

So the angle at the edge is half of that: 90°, always.

Any triangle drawn on a diameter is right-angled. That is a special case, not a new rule, and it is enormously useful.

Opposite corners of a cyclic quadrilateral

If all four corners of a quadrilateral sit on a circle, its opposite angles add to 180°.

The reason is the doubling rule again: the two centre angles on either side together make a full 360°, so the two edge angles make half of that.

Equal sides are angles in disguise

The moment you see two equal sides, you have two equal angles for free.

That is why a diagram full of tick marks is a gift: each pair of ticks is a pair of angles you have not been told about but already know.

Mark them in first, before you try to work anything out.

Two triangles sharing a side

Chains happen when an isosceles triangle shares a side with another.

Work out the angles of the first, then carry one of them into the second as a known value.

The answer usually appears three or four steps in — which is exactly why writing each angle onto the picture matters so much.

Radius meets tangent at ninety degrees

Wherever a tangent touches, the radius drawn to that point meets it at a right angle.

So any tangent in a diagram hands you a 90° angle you were not given. That is usually the way in.

And two tangents drawn from the same outside point are always equal in length — which makes an isosceles triangle appear as well.

Two tangents make a kite

Draw both tangents from an outside point, plus the two radii. The four lines make a kite: two pairs of equal sides meeting at a corner.

The two right angles sit opposite each other, so they use up 180° of the kite’s 360°.

That leaves the angle at the centre and the angle at the point adding to 180° — always.

Opposite angles add to a straight line

A cyclic quadrilateral has all four corners on one circle. Its opposite angles always add to 180°.

So knowing one angle gives you the one across from it immediately, and knowing two adjacent ones gives you all four.

And it works the other way: if opposite angles add to 180°, the four points must lie on a circle.

The exterior angle equals the opposite interior one

Extend one side of a cyclic quadrilateral. The exterior angle you make is equal to the interior angle at the opposite corner.

That follows straight from the 180° rule: the exterior angle is 180 - its neighbour, and so is the opposite angle.

It looks like a new theorem and it is the old one rewritten.

When two shapes share an edge

Put a square and an equilateral triangle side by side on the same line, sharing an edge.

The square gives 90°, the triangle gives 60°, and a full turn is 360°. So the gap between them is 360 - 90 - 60 = 210°, or 150° the other way.

Most "find the angle between two shapes" puzzles are exactly this: add up what you know and take it from 360.

Working out what fits

Three regular polygons meet at a point with no gap. What could they be?

Their angles must add to exactly 360°. Three hexagons: 120 × 3 = 360. Works.

A square, a hexagon and a twelve-sided shape: 90 + 120 + 150 = 360. Also works. Tiling puzzles are just this equation.