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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.

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Polygon Cradle

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?

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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: Challenge Level:2 Challenge Level:2

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.