Chapter 6.1, Problem 51HP

To determine

To find: The reason that when two sides of the regular hexagon are extended to meet at a point in the exterior of the polygon, the triangle formed are equilateral.

Answer to Problem 51HP

The triangle formed when two sides of the regular hexagon are extended to meet at a point in the exterior of the polygon are equilateral.

Explanation of Solution

Consider a hexagon

  

Now, extend the side PV and ST.

Let is meets the points at U.

  

Here the interior angle $\angle PVT=\angle STV=120°$ .

Thus, there exterior angle $\angle UVT\text{and}\angle UTV$ also be equal.

Therefore, since. $\angle UVT=\angle UTV$

Thus, $UV=UT$ .

Hence, the triangle formed when two sides of the regular hexagon are extended to meet at a point in the exterior of the polygon are equilateral.