Question
ABCD is a cyclic trapezium with AD || BC. If $\angle\text{B}=70^\circ,$determine other three angles of the trapezium.
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Answer
We have
ABCD is a cyclic trapezium with AD || BC and $\angle\text{B}=70^\circ.$
Since, ABCD is a cyclic quadrilateral
Then, $\angle\text{B}+\angle\text{D}=180^\circ$
$\Rightarrow70^\circ+\angle\text{D}=180^\circ$
$\Rightarrow\angle\text{D}=180^\circ-70^\circ=110^\circ$
Since, AD || BC
Then, $\angle\text{A}+\angle\text{B}=180^\circ$ [Co-interior angles]
$\Rightarrow\angle\text{A}+70^\circ=180^\circ$
$\Rightarrow\angle\text{A}=180^\circ-70^\circ=110^\circ$
Since, ABCD is a cyclic quadrilateral
Then, $\angle\text{A}+\angle\text{C}=180^\circ$
$\Rightarrow110^\circ+\angle\text{C}=180^\circ$
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