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At a Glance

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

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No Right Angle Here

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

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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

Triangle Split

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Since $SP = SQ$, the triangle $PSQ$ is isosceles. Therefore, $\angle SPQ = \angle PQS$. We denote the measure of those angles by $y$. Similarly, $\angle SQR = \angle QRS = x^\circ$.

Since the sums of the interior angles of $PQR$ is $180^\circ$, $x + y + (x+y) = 180$, so $ 2x+2y=180 $.

Therefore, $ \angle PQR = x^\circ+y^\circ=90^\circ $.

This problem is taken from the UKMT Mathematical Challenges.
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