2835
/www/nrich/html/content/id/2835/
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2011-02-01T00:00:01
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The diagram, which is not drawn accurately, shows a parallelogram inside a triangle.The marked lengths are equal.<br></br>
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What is the value of $x$?<br></br>
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<mdo:image alt="triangle with parallelogram" height="117" src="wp27.JPG" width="295"></mdo:image><br></br>
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If you liked this problem, <a href="http://nrich.maths.org/public/viewer.php?obj_id=2927&refpage=titlesearch.php">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br></mdoxml>
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<mdo:image height="118" width="295" src="wp27s.JPG" alt="diagram illustrating angles"></mdo:image><br></br>
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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 $x =90$. <br></br></mdoxml>
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Weekly Problem 27 - 2013
Weekly Problem 27 - 2013
2D Geometry, Shape and Space
Parallelograms
2D Geometry, Shape and Space
Mixed triangles
2D Geometry, Shape and Space
Angle properties of shapes