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# A villager Ram has a plot of land in the shape of a concave quadrilateral ABDFA as shown in the figure below. He himself decided to construct a health care center for the villagers in the area FCD and keep the remaining with him.(i) Find the value of x.(ii) What values of Ram are depicted while deciding to do so?

Last updated date: 02nd Aug 2024
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Hint: Sum of the interior angles of a triangle is 180˚.
Concave quadrilaterals are the quadrilaterals which have at least one interior angle greater than 1800.

(i) Since, the sum of the three angles of a triangle is 180˚, we can say that in Δ ABC:
∠ A + ∠ B + ∠ C = 180˚
On putting values of angle A and angle C,
⇒ 40˚ + ∠ B + 90˚ = 180˚
On simplifying,
⇒ ∠ B = 180˚ - 40˚ - 90˚ = 50˚

Now, in Δ BDE:
∠ B + ∠ D + ∠ E = 180˚
On putting values,
⇒ 50˚ + x˚ + 100˚ = 180˚
⇒ x˚ = 180˚ - 50˚ - 100˚ = 30˚
(ii) By giving a part of his land for a health care center for the villagers, Ram depicts the qualities of generosity, responsibility, optimism etc.

Note: In general, the sum of the exterior angles of any polygon is always 360˚.
Since, there are n angles in an n-sided polygon and the sum of an interior angle and its exterior angle is 180˚, we may calculate that the sum of the interior angles will be (180n - 360)˚.
For a triangle, the sum of the interior angles is always: 180 × 3 - 360 = 540 - 360 = 180˚.
For a quadrilateral, the sum of the internal angles is always: 180 × 4 - 360 = 720 - 360 = 360˚.