Question

Problem 2 (10 points) (a) In Euclidean geometry explain why the sum of the angles of a triangle equals two right angle. (b) In spherical geometry, what can you say about the sum of the angles of a spherical triangle?
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Answer 1

Solution. (@2 Euclidon. Geomeby, the study of plane and solid figures on the basis of axioms and theorems employed by the 677.uk Mathematian. EUCLID fundamental that the sum is 180° or The parallel postulate is for the proof of the theorem of the angles cha triangle Sum of two Right angles. figure. to / Draw D6 parallet to AB A transversal throught two parallel lines LDE and AB), the angles formed suck as a B (Altenate angles and the three angles & B and r on The line De must sum to 180° ( Angles sum on a Straight line is alueys 180) de x B = B and r is common. So. Sum of the angles of a triangle is 180 - ě qo' t o = 180 (Sum of 2 Right angles/

A spherical thangle is a figure formed on the surface of a sphere by three great Circular arcs intersecting beurwise in theree vertices. Let a spherical triangle have angles A, B and C (Measured in redean's ) and let the sphere on which the Spherical triangle sits have Radius=R. The sum of the angles of a spherical trangle is between I and 3 Tiradians (180 and 540 The amount by which it enceeds 180' is s enical excess and I denoted by E.