Chapter 12.7, Problem 25HP

To determine

Comparing the sum of the angles measures of a triangle in hyperbolic, spherical and Euclidean geometries.

Answer to Problem 25HP

In Euclidean geometry sum of angle is ${180}^{\circ }$ , in spherical the sum is greater than ${180}^{\circ }$ and in hyperbolic less than ${180}^{\circ }$ .

Explanation of Solution

Given:

It is given in the question that the figure is shown below:

  

Concept Used:

In this, use the concept of knowing that geometries can be defined on curved surfaces other than spheres and the hyperbolic geometries is defined on a curved saddle like surface.

Since, In Euclidean geometry (traditional plane geometry), the angles in a triangle will be the sum to ${180}^{\circ }$ . In spherical geometry , the angles sum is more than ${180}^{\circ }$ ( It is possible with three right angles) and In hyperbolic geometry, the angles will sum to the less than ${180}^{\circ }$ .