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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

Equal Lengths

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

diagram of triangle inside rhombus with illustrated angles
Triangle $DFC$ is isosceles $(CF=CD)$.

Hence $\angle DFC = \angle FDC = x^{\circ}$.

Hence $\angle FCD = (180-2x)^{\circ}$ (angle sum of triangle).

Now $\angle EBC = \angle FDC = x^{\circ}$ (opposite angles of a parallelogram) and triangle EBC is isosceles (CE = CB). Hence $\angle BEC = x^{\circ}$ and $\angle ECB = (180-2x)^{\circ}$

Lines $AD$ and $BC$ are parallel and hence $$\angle ADC + \angle BCD = 180^{\circ}$$ Therefore: $$x+2(180-2x) + 60= 180$$i.e.$420-3x=180$ i.e. $3x=240$ i.e. $x=80$

This problem is taken from the UKMT Mathematical Challenges.

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