### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

### A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

# Trapezium Arch

##### Stage: 3 and 4 Short Challenge Level:

Consider the ten trapezia. Add an eleventh trapezium to one edge of the arch.
To move from the left of the arch to the right, each trapezium has to turn the same number of degrees.

Consider the blue edgeof the bottom left hand stone. To complete the arch it has to turn a total of $180$ degrees in 10 steps. That is $18$ degrees for each step, or $9$ degrees for each of the base angles of each trapezium.

Therefore the largest angles of each trapezium are $(90+9)$ degrees and the smallest $(90-9)$ degrees = 81 degrees .

This problem is taken from the UKMT Mathematical Challenges.
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