Shared Vertex
Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?
In the diagram, what is the value of $x$?
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![Shared Vertex Shared Vertex](/sites/default/files/styles/large/public/thumbnails/content-id-11721-Weekly%2525202017%252520-%25252038%252520Diagram%2525202.png?itok=soWkYEq0)
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
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![Shared Vertex Shared Vertex](/sites/default/files/styles/large/public/thumbnails/content-id-11721-Weekly%2525202017%252520-%25252038%252520Solution%2525202.png?itok=ATOFWD4m)
Angles in a triangle add up to $180^\circ$, so applying this in $BCD$ gives:
$\angle CBD = 180^\circ - \angle BCD - \angle BDC = 180^\circ - 65^\circ - 65^\circ = 50^\circ$.
$\angle CBD$ and $\angle ABE$ are opposite angles at $B$, so are equal. Therefore $\angle ABE = 50^\circ$.
Since the angles in triangle $ABE$ add up to $180^\circ$, $x = 180^\circ - \angle BEA - \angle ABE = 180^\circ - 90^\circ - 50^\circ = 40^\circ$.