### Bow Tie

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

Weekly Problem 18 - 2007
A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?

### Hexapentagon

Weekly Problem 53 - 2007
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?

# U in a Pentagon

##### Stage: 3 Short Challenge Level:

Each interior angle of a regular pentagon is $108^\circ$, so $\angle SRQ=108^\circ$. As $SR=QR$, the triangle is isosceles with $\angle RQS=\angle RSQ = 36^\circ$. Similarly, $\angle SRT= \angle STR = 36^\circ$. So $\angle SUR=(180-2\times36)^\circ=108^\circ$. From the symmetry of the figure, $\angle PUR=\angle PUS= (360^\circ - 108^\circ)/2 = 126^\circ$.

This problem is taken from the UKMT Mathematical Challenges.