.Use the information given in the following figure to find :
$(1)$ x
$(2)$ $∠ B$ and $∠ C$

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Solution

Given angles are $∠ A = 90^{\circ}, ∠ B = 2 x + 4^{\circ}, ∠ C = 3 x - 5^{\circ}, ∠ D = 8 x - 15^{\circ}$
In quadrilateral $A B C D$
1) $∠ A + ∠ B + ∠ C + ∠ D = 360^{\circ}$
$90^{\circ} + 2 x + 4^{\circ} + 3 x - 5^{\circ} + 8 x - 15^{\circ} = 360^{\circ}$
$13 x = 286^{\circ} ⟹ x = 22^{\circ}$
$2) ∠ B = 2 x + 4^{\circ} = 2 (22^{\circ}) + 4 = 48^{\circ}, ∠ C = 3 x - 5^{\circ} = 3 (22^{\circ}) - 5^{\circ} = 61^{\circ}$

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