Читать книгу The Little Book of Mathematical Principles, Theories & Things - Robert Solomon - Страница 27

The sum of the angles in a triangle is two right angles, or 180°.

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For any triangle, the sum of its angles is always 180°. This result relies on a disputed axiom of geometry and is familiar from school geometry.

Take a triangle ABC and draw a line DAE parallel to BC.

Then ∠ABC = ∠DAB and ∠ACB = ∠EAC.

So ∠BAC + ∠ABC + ∠ACB = ∠BAC + ∠DAB + ∠EAC.

The left-hand side of this equation, ∠BAC + ∠ABC + ∠ACB, is the sum of the angles of the triangle.

The right-hand side, ∠BAC + ∠DAB + ∠EAC, is the sum of the angles along a straight line, which is 180°. So the sum of the angles of the triangle is 180°.

In the proof above we stated that:

∠ABC = ∠DAB and ∠ACB = ∠EAC

These pairs of angles are known as “alternate angles”. The equality of alternate angles is a consequence of the fifth postulate of Euclid concerning parallel lines. A different postulate could give a different result: it might be that the sum of the angles is less than 180°, or greater than 180°.

Summing the angles.