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Numbers and Math · entry 33 of 52
Triangles always have interior angles that add to one hundred eighty degrees in flat Euclidean geometry.
The longer version
Draw a line through one corner parallel to the opposite side, and the two base angles reappear on either side of the top corner as alternate interior angles. Together with the angle already sitting there, they fill a straight line, which measures one hundred eighty degrees.
The proof leans on a guarantee about parallel lines, and that guarantee is an assumption rather than a consequence of the other rules of geometry. Drop it and the total changes. Surveyors rely on the flat result constantly, adding angles to check their measurements, and any mismatch signals error rather than a curved surface.
Where this entry sits
This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.
More From Numbers and Math
Eight more entriesThe angles of a triangle drawn on a sphere add to more than one hundred eighty degrees, unlike flat triangles.
Read 35Parallel lines never meet in flat geometry, but on a curved surface they can eventually converge.
Read 36The Pythagorean theorem says the square of the hypotenuse equals the sum of the squares of the other sides.
Read 37A three, four, five triangle is the simplest example of a right triangle with whole number side lengths.
Read 38The number six is perfect because its proper divisors of one, two, and three add up to six.
Read 39Amicable numbers come in pairs where the proper divisors of each sum to the other, as with two hundred twenty and two hundred eighty-four.
Read 40A magic square arranges numbers so that every row, column, and diagonal adds to the same total.
Read 41The birthday problem shows that a room of roughly twenty-three people has better than even odds of a shared birthday.
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