In the figure, what is y in terms of x?

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Solution

In $Δ A B C,$

$∠ A C B = 180^{\circ} - (x + 2 x)$

= $180^{\circ} - 3 x$ ...(i)

and in $Δ B D G,$

$∠ B E D = 180^{\circ} - (2 x + y)$ ...(ii)

$∠ E G C = ∠ A G D$

(Vertivally opposite angles)

= 3y

In quad. BCGE,

$∠ B + ∠ A C B + ∠ C G E + ∠ B E D = 360^{\circ}$

(Sum of angles of a quadrilateral)

$\Rightarrow 2 x + 180^{\circ} - 3 x + 3 y + 180^{\circ} - 2 x - y = 360^{\circ}$

$\Rightarrow - 3 x + 2 y = 0$

$\Rightarrow 3 x + 2 y \Rightarrow y = \frac{3}{2} x$

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