# Inscribed Hexagon

Weekly Problem 1 - 2014

The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?

The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?

## Problem

Image

The diagram shows a regular hexagon inside a rectangle.

What is the sum of the four marked angles?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

## Student Solutions

As the sum of the angles in a triangle is 180 ° and all four angles in a rectangle are 90 °, then sum of the two marked angles in the triangle 180 ° - 90 ° = 90 °.

Each interior angle of a regular hexagon is 120 ° and the sum of the angles in a quadrilateral is 360 °; hence the sum of the two marked angles in the quadrilateral is 360 ° - 90 ° - (360 ° - 120 °) = 30 °.

Hence the sum of the four marked angles is 90 ° + 30 ° = 120 °.