Question
$\text{ABCDE}$ is a pentagon in which $AB$ is parallel to $DC$ and $\angle A : \angle E : \angle D = 1 : 2 : 3.$ Find angle $A.$
✓
Answer
Given $AB \| DC$
$\Rightarrow \angle B + \angle C = 180^\circ $
Also,
$\angle A : \angle E : \angle D = 1 : 2 : 3$
$\Rightarrow \angle A = x, \angle E = 2x$ and $\angle D = 3x$
Since,
$\angle A + \angle B + \angle C + \angle D + \angle E = (5 - 2) \times 180^\circ $
$\Rightarrow \angle A + (\angle B + \angle C) + \angle D + \angle E = 3 \times 180^\circ $
$\Rightarrow x + 180^\circ + 3x + 2x = 540^\circ $
$\Rightarrow 6x + 180^\circ = 540^\circ $
$\Rightarrow 6x = 360^\circ $
$\Rightarrow x = 60^\circ $
Hence,
$\angle A = 60^\circ .$
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