# Find the angle x

#### Farzin

If it can be drawn with those angles, they must be correct, as only one correct diagram is possible. Note that angle ADE is determined by the remaining angles shown - it has to be $108^\circ$.
IF it can be drawn!

#### skipjack

Forum Staff
Below is an expanded diagram that should help.

#### Farzin

Below is an expanded diagram that should help.View attachment 11015
Apart from a few obvious angles the rest are not so obvious and there is no way to prove their measurements. Of course they are measurable by geometry software but how about a real solution on the paper? How do we get those 54° and 60° ?

#### skipjack

Forum Staff
Draw the pentagon, equilateral triangle ABG and line segment BF first, then draw CB equal in length to DF with angle ABC equal to 96$^\circ$ and C to the left of B. Add the label E as shown. Now draw CD. The symmetry of the pentagon about the line segment BF implies the angles shown at B, F and E. As EB = EF (they are opposite equal angles of triangle BEF), CE = DE, and so angles ECD and EDC each equal 54$^\circ$.

The above proves that the diagram is possible, so the initial guess that angle ADF is 108$^\circ$ is correct.

#### Farzin

Draw the pentagon, equilateral triangle ABG and line segment BF first, then draw CB equal in length to DF with angle ABC equal to 96$^\circ$ and C to the left of B. Add the label E as shown. Now draw CD. The symmetry of the pentagon about the line segment BF implies the angles shown at B, F and E. As EB = EF (they are opposite equal angles of triangle BEF), CE = DE, and so angles ECD and EDC each equal 54$^\circ$.

The above proves that the diagram is possible, so the initial guess that angle ADF is 108$^\circ$ is correct.
Excellent job, Very well done. Thank you very much.