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Parallelogram in the Middle

Age 11 to 14 Short
Challenge Level

                     

The two angles marked $y ^{\circ}$ are equal because they are in an isosceles triangle. For the same reason, the angles $z^{\circ}$are equal. Since an exterior angle of a triangle is the sum of the two interior and opposite angles, it follows that $a=2y$ and $b=2z$. Now $a^{\circ}+ b^{\circ} = 180^{\circ}$ since they are the base angles of a parallelogram. So $2y + 2z = 180$ giving $y+z=90$. But, from the angle sum of a triangle $x+y+z=180$; hence $x =90$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.