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Weekly Problem 8 - 2016
Can you work out the size of the angles in a quadrilateral?
Can you work out the size of the angles in a quadrilateral?
Age
11 to 14
Challenge level
Problem
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The diagram shows a quadrilateral $ABCD$, in which $AD=BC$, $\angle CAD = 50^\circ$, $\angle ACD = 65^\circ$ and $\angle ACB = 70^\circ$.
What is the size of $\angle ABC$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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This means that $\angle ACD = \angle CDA$, so $ACD$ is an isosceles triangle. Therefore, $AC = AD$.
We know from the question that, $AD = BC$, so $BC = AC$. This makes $ABC$ an isosceles triangle, and $\angle CAB = \angle ABC$.
Then $\angle ABC = \frac{1}{2} \left( 180^\circ - \angle ACB \right) =\frac{1}{2} \left( 180^\circ - 70^\circ \right) = 55^\circ$.