Angle Chase

find the marked angle, one named rule at a time

Worked example — Angle Chase

Steps in the chain	2
Rules used	2
Pinned down	5 / 6
Other ways to this angle	6, from 1 to 3 steps
Given	30°

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Angle Chase

An angle chase is the exercise of finding an unmarked angle in a figure by applying one named rule at a time, each step using angles you have already found. The rules are the small fixed set a geometry course gives you, and every one of them is a linear equation in the unknown angles: the angle sum of a triangle is a + b + c = 180, vertical angles are a = b, and so on. That is why elimination can always answer a chase, and why the interesting question is not what the angle is but whether the figure has told you enough to find it, and how far you have to walk once it has.

What each symbol means

180°: the angle sum of any triangle, and of the angles filling one side of a straight line
360°: the angles round a point
rank: how many of the figure's unknown angles the givens and rules actually pin down

Assumes: A plane figure, drawn without contradiction. Every figure here is built by choosing its angles first and then realising the coordinates, so it cannot be impossible by accident — a guarantee a hand-drawn textbook figure does not come with.

You can answer the question without the figure being solved 🖖

Click One step and read the card marked Pinned down. It says 4 of 6. The figure holds six unknown angles, and what you were given, together with the rules, nails down four of them. Two stay free: the cevian could meet BC anywhere along a range and nothing you were told would change. The angle you were asked for is certain regardless. That gap is worth sitting with, because being stuck feels identical in both cases. A reader who cannot find ∠ADB is missing a step. A reader who cannot find ∠BAD is missing nothing at all, because that angle has no value here to find. Counting unknowns against what the rules pin down separates the two in one line, and answers a question that searching never does: whether there is an answer at all.

Nine rules, and repeating one is what makes a chase long 🖖

Every chain this tool can produce is built from nine rules: the angle sums of a triangle and of a quadrilateral, angles on a straight line, vertical angles, angles round a point, the equal base angles of an isosceles, alternate and corresponding angles across a pair of parallels, and adding the parts of a split angle back together. On seven of the eight figures a chase hardly ever uses one of them twice — the quadrilateral never does, the crossing does in 0.0% of its chases, the triangle cut into three in 4.7%. Then there is the staircase, where 30.1% of chases use some rule three times or more. It is also the only figure that reaches eight steps, and those two facts are the same fact: it gets long by repeating itself, carrying one value from one end of the path to the other, alternate angle then straight line, bend after bend. Length comes from distance, not from difficulty.

A long chase is usually the question, not you 🖖

Press Another puzzle a few times without touching the figure. The angle being asked for often stays the same; what changes is which angles you were handed. On the staircase, ∠A0A1A2 can be reached 32 different ways, and the shortest is one step while the longest is eight. Same figure, same angle, same answer of 30°. Across all eight figures here, 63.0% of angles can be reached by chases that differ by two steps or more, and 32.8% by three or more. So when a chase drags on, that is usually a fact about which angles the question gave you rather than about how you are doing it, and the card marked Other ways to this angle prints both numbers for whatever is on screen. It is also why comparing your working with somebody else’s is worth doing and worth not worrying about: two correct chases of very different lengths are the ordinary case.

Problem solved in full

One step, or eight, for the same angle 5 steps
In the staircase, ∠A0A1A2 is 30°. Find it twice — once from the angles that make it easy, once from the angles that do not — and say what the difference is a fact about. Choose Staircase between parallels and press Another puzzle until the question asks for ∠A0A1A2.

Open this problem in the tool
1. The answer does not depend on any of this. ∠A0A1A2 is 30° in this figure and would be 30° whatever you were told, because the figure was built before the question was asked.
  
  $∠ A_{0} A_{1} A_{2} = 3 0^{\circ}$
2. Here is the easy way in. If the question hands you the two angles either side of it at A1, the whole chase is one subtraction on a straight line: the three angles at A1 below the lower parallel fill 180°, and two of them are known.
  
  $from ∠ N A_{1} A_{0} = 3 0^{\circ} and ∠ A_{2} A_{1} M = 12 0^{\circ} : 18 0^{\circ} - 3 0^{\circ} - 12 0^{\circ} = 3 0^{\circ}$
3. Now the hard way in, with the same angle asked for. You are given one angle at A0, one at A2 and one at A3 — nothing at A1 at all. Every one of them has to be walked along the path to reach A1, and that takes eight steps, three of which cross between the parallels.
  
  $from ∠ A_{1} A_{0} R, ∠ A_{3} A_{2} A_{1}, ∠ A_{2} A_{3} M : eight steps, three of them across the parallels$
4. Neither route is cleverer than the other. What separates them is only which angles the question chose to mark, and this figure offers 32 different choices for the same target, the shortest one step and the longest eight.
  
  $what the question handed you both angles at A_{1} one angle at each of A_{0}, A_{2}, A_{3} steps 18$
5. That spread is not unusual. Across the eight figures here, 63.0% of angles can be reached by chases that differ by two steps or more, and 32.8% by three or more.
  
  $32 ways in, 1 \leq steps \leq 8$
Answer

The length of a chase is a property of the question, not of the person doing it. This is worth knowing for a practical reason: when a chase runs long, the useful response is to check whether a different angle in the figure is easier to reach first, not to assume you have missed a shortcut. It is also why two correct solutions of very different lengths are the ordinary case rather than a sign that one of them is wrong — and why an examiner marking the working, rather than the number, is marking the harder thing.

$same figure, same angle, same 3 0^{\circ}$

References (3)

The angle sum, which is the rule every chain on this page leans on hardest: Euclid, Elements, Book I, Proposition 32. D. E. Joyce (ed.), Clark University.
Vertical angles, the rule that is the only bridge between the two halves of the crossing: Euclid, Elements, Book I, Proposition 15. D. E. Joyce (ed.), Clark University.
The equal base angles, which is where the chase stops and the algebra starts: Euclid, Elements, Book I, Proposition 5 (the pons asinorum). D. E. Joyce (ed.), Clark University.

By the 42Math team · Reviewed against published sources. Corrections welcome.