# Parallelogram in the Middle

Weekly Problem 27 - 2013

The diagram shows a parallelogram inside a triangle. What is the value of $x$?

The diagram shows a parallelogram inside a triangle. What is the value of $x$?

## Problem

The diagram, which is not drawn accurately, shows a parallelogram inside a triangle.The marked lengths are equal.

What is the value of $x$?

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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

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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 $x =90$.