Chapter 33, Problem 20P

(a)

To determine

To describe: A triangle for the given sum of the angles.

Explanation of Solution

Introduction:

Sum of all the interior angles of a triangle is $180°$ .

As shown in the figure below, a triangle is drawn on the sphere. Considering one vertex on the North Pole and the other two vertices on the latitude of the sphere. Such that, the other two angles will be a right angle, and due to the curvature of the sphere, the lines of the triangle will be curved outwards. This makes the third angle of $179°$

  

If such a triangle is formed on the curvature of the sphere, then the sum of the limiting angles would be $359°$ .

Conclusion:

A triangle is drawn whose two vertices on the latitude lines and the third one on the North Pole.

(b)

To determine

To describe: A triangle for the given sum of the angles.

Explanation of Solution

Introduction:

Sum of all the angles in a triangle is $180°$.

If a triangle is drawn on the surface of a sphere, then due to the curvature of the sphere, the sum of angles would be greater than $180°$ up to the limiting value of the sum $359°$ . Now, if that triangle is reduced to such a small extent that lines of the triangle on the curvature of the sphere becomes straight. In that case, the triangle would behave like an equilateral triangle whose sum of all the angles inside the triangle will be $180°$ .

Conclusion:

Thus, a triangle drawn on a flat surface has the sum of $180°$ .