Angle Chasing

Angle-chasing is the detective game of geometry. You are handed a figure crowded with lines and triangles and just one or two known angles. From those tiny clues you deduce every other angle in the picture — one at a time, each new angle unlocking the next, like following footprints across a room until you catch the culprit.

The best part: you never measure and you never guess. Each step is forced by an angle rule you already know — alternate angles, co-interior angles, angles on a line, the angles of a triangle. This page is about doing that: first watch a chase, then run some yourself.

The detective's toolkit

Every deduction in a chase leans on one of these facts. Keep them at your fingertips — this is your whole kit of clues:

Learn to spot the shapes — the X, F, Z, C and the triangle — and each one tells you what to do next.

Watch one chase

Two parallel lines (note the matching arrows) are cut by a transversal. We are told one angle is 65^\circ. Step through to see how that single fact forces two more.

Read it as a chain, each link naming its rule: 65^\circ given \Rightarrow alternate angles give another 65^\circ \Rightarrow angles on a line give 180^\circ - 65^\circ = 115^\circ. Three angles known, from one clue.

A second chase: spot the isosceles triangle

Not every chase is about parallel lines. Here a triangle has two equal sides (watch for the matching tick marks). The apex angle at A is 40^\circ; find the base angles. The trick is noticing the triangle is isosceles — then its two base angles must be equal, and the rest is arithmetic.

The chain: the three angles add to 180^\circ, the apex takes 40^\circ, so the two equal base angles share what is left:

\frac{180^\circ - 40^\circ}{2} = \frac{140^\circ}{2} = 70^\circ \text{ each}.

Had we instead been given a base angle of, say, 70^\circ, we could chase the other way: the base angles are equal (another 70^\circ), so the apex is 180^\circ - 70^\circ - 70^\circ = 40^\circ.

The marks of the trade

A good angle-chase diagram is covered in marks that record what you know: ticks for equal lengths, arcs for equal angles, a small square for a right angle (and the matching arrows for parallel lines you saw above). Here is an isosceles triangle with its altitude — the two equal sides are ticked, the two equal base angles are arced, and the square marks where the altitude meets the base at a right angle.

In angle-chasing, how you know is as important as what you know. Every step must be justified with the specific rule you used — not "it's obviously 65°" but "alternate angles, parallel lines". A bare number with no reasoning throws away marks in an exam, and worse, it means you can't tell whether you were right or just lucky.

And never measure or eyeball. Diagrams are almost always drawn "not to scale" on purpose. An angle that looks about 60^\circ might really be 72^\circ. Guessing from the picture is not maths — deducing from a rule is. If you cannot name the rule, you have not finished the step.

Angle-chasing is exactly the skill behind competition geometry — the fiendish figures at Maths Olympiads around the world. A problem can look hopelessly tangled, a mess of circles, triangles and crossing lines, and then a clever chain of angle deductions makes a stunningly simple answer fall out, as if by magic. That "aha!" — where a long chase suddenly closes on a neat number like 90^\circ — is why people find geometry genuinely addictive.

It is geometry as a puzzle sport: no formula to memorise, just clues, rules, and the thrill of the hunt. The more chases you run, the faster you spot the shapes — and the better a detective you become.

Solve one yourself

Here is a freshly generated chase. Some angles are given; the highlighted box is the one to find, with a step or two of theorem in between. Type each unknown angle, use the tools to mark parallels or draw a helper line if it aids your thinking, then press Check. Hit Refresh for a brand-new figure — the problems are built only from the theorems taught in the lessons leading here.