Find the Missing Angle: Geometry Puzzles to Solve

Angle puzzles as a brain game

“Find the missing angle” is a school worksheet staple — and, it turns out, a great puzzle for adults. Each problem is a small piece of deduction: you know a few facts about lines and triangles, you are given two numbers, and you chain the facts together to find x. The Geometry game in Memo turns these into a six-puzzle round: two easy, three medium and a two-step boss.

The eight rules that solve almost everything

These are the same hints the game gives you when you are stuck.

1. Triangle sum: the three inside angles of any triangle add up to 180°.
2. Straight line: angles side by side on a straight line add up to 180°.
3. Vertical angles: when two straight lines cross, opposite angles are equal.
4. Exterior angle: an outside angle of a triangle equals the sum of the two opposite inside angles.
5. Isosceles triangle: two equal sides (marked with tick lines) mean the two angles opposite them are equal.
6. Equilateral triangle: all sides equal means every angle is 60°.
7. Right triangle: the 90° corner takes half of 180°, so the other two angles add up to 90°.
8. Pythagoras: in a right triangle, a² + b² = c², where c is the side opposite the right angle.

Where the missing-angle rules come from

Most of these rules are more than 2,000 years old. They are proved in Book I of Euclid’s Elements, a 13-book textbook written by a Greek mathematician who taught in Alexandria and probably lived from about 325 to 265 BC. As the MacTutor archive notes, Euclid probably proved few of the results first; his genius was the organization. In Book I:

• Proposition 13: angles on a straight line make “two right angles” — Euclid’s way of saying 180°.
• Proposition 15: vertical angles are equal.
• Proposition 32: the exterior angle rule and the triangle sum, proved together.
• Proposition 5: the base angles of an isosceles triangle are equal. It is nicknamed the pons asinorum, or “bridge of asses”; nobody is sure whether that mocks the students who got stuck on it or just describes its bridge-like diagram.
• Proposition 47: the Pythagorean theorem.

That last one is older than its name. Pythagoras lived from about 570 to 490 BCE, and the Stanford Encyclopedia of Philosophy notes that he probably did not prove the theorem. The Babylonians knew the rule long before him. A clay tablet called Plimpton 322, written between the 19th and 17th centuries BCE and now at Columbia University, shows that scribes had a method for generating Pythagorean triples about a millennium and a half before Euclid. Scholars still argue about whether it was a teaching exercise or something closer to research.

The intuition may be older still. In a 2011 study, researchers asked adults and children of the Mundurucu, an Amazonian group, about triangles on a surface described as perfectly flat. Without any formal math education, their estimates of the inside angles added up to about 180°, in line with schooled adults and children in the US and France.

Worked examples

1. Triangle: two angles given

A triangle has angles of 52° and 71°. The third angle is 180° − 52° − 71° = 57°.

2. Straight line

A ray meets a straight line, making 125° on one side. The angle on the other side is 180° − 125° = 55°.

3. Exterior angle

A triangle’s inside angles at the other two corners are 48° and 63°. The exterior angle at the third corner is 48° + 63° = 111° — no need to find the inside angle first.

4. Isosceles triangle

The top angle of an isosceles triangle is 40°. The two base angles are equal, so each is (180° − 40°) ÷ 2 = 70°.

5. The boss: two steps

An isosceles triangle has its base extended, and the outside angle at the base is 115°. Step 1, straight line: the base angle is 180° − 115° = 65°. Step 2, isosceles: the other base angle is also 65°. Step 3, triangle sum: the top angle is 180° − 65° − 65° = 50°.

6. Pythagoras

A right triangle has legs of 6 and 8. The long side is √(6² + 8²) = √100 = 10. Common whole-number triples worth knowing: 3-4-5, 5-12-13, 8-15-17, and their multiples.

Why geometry puzzles make a good daily brain game

Unlike speed games, angle puzzles reward slowing down and reasoning step by step. They are a gentle workout for spatial thinking and logical deduction, and the satisfaction of a solved two-step problem is real.

Does it improve your brain? What the research says

We know of no study of angle puzzles as brain training, but there is research on the skills they use, and on brain games in general.

• Spatial skills respond to practice. A meta-analysis of 217 studies found an average improvement of about half a standard deviation over control groups. The gains held up when testing was delayed and carried over to untrained spatial tasks. That covers spatial training in general, not angle puzzles, so it is a reason for optimism rather than proof.
• Math has its own brain network. In a 2016 brain-imaging study, 15 professional mathematicians and 15 non-mathematicians judged math and non-math statements. In the mathematicians, statements in algebra, analysis, topology and geometry all lit up the same frontal, parietal and temporal regions, and spared areas used for language. The authors conclude that advanced math grows out of the same nonverbal circuits as basic number sense.
• Brain games mostly train themselves. In a BBC-run study, 11,430 adults practiced reasoning, visuospatial and memory tasks for six weeks. They improved on every trained task, but not on untrained tests, even closely related ones.
• The overall picture. A 2016 review of the studies cited by brain-training companies found extensive evidence for gains on trained tasks, less for closely related tasks, and little for everyday thinking.

So expect to get better at angle puzzles, and enjoy the deduction for its own sake; broader claims about boosting intelligence aren’t backed by solid evidence. For a faster kind of numbers game, try the mental math game.