Chapter 7, Problem 17RP

To determine

To prove : The congruency of the diagonals of a regular pentagon

Explanation of Solution

Given information : The following information has been given

Regular pentagon

All angles are congruent

All sides are congruent

Formula used : Any two triangles are said to be congruent by SAS property if they have two congruent sides, and the angle containing then is also congruent

Proof : We know that as the pentagon is of sides, any diagonal will have only one vertex on one side of the diagonal, and only two vertices of pentagon on the other. There is no other possibility.

Thus, we assume a triangle with diagonal as one side and sides of pentagon as the other two.

We know that in a pentagon

All angles are congruent

All sides are congruent

Hence, we can say that all the triangles containing a diagonal will be congruent by SAS property.

Now, by virtue of congruency, we can say that the diagonal will be congruent always, for any sides it’s chosen for.

Hence, proven that the diagonals are congruent