Q. 173.8( 5 Votes )

Find the angles of a cyclic quadrilateral ABCD in which
∠A = (4x + 20)°, ∠B = (3x - 5)°, ∠C = (4y)° and ∠D = (7y + 5)°.

Answer :

It is given that angles of a cyclic quadrilateral ABCD are given by:


∠A = (4x + 20)°,


∠B = (3x - 5)°,


∠C = (4y)°


and ∠D = (7y + 5)°.


We know that the opposite angles of a cyclic quadrilateral are supplementary.


∠A + ∠C = 180°


4x + 20 + 4y = 180°


4x + 4y – 160 = 0 … (1)


And ∠B + ∠D = 180°


3x – 5 + 7y + 5 = 180°


3x + 7y - 180° = 0… (2)


By elimination method,


Step 1: Multiply equation (1) by 3 and equation (2) by 4 to make the coefficients of x equal.


Then, we get the equations as:


12x + 12y = 480 … (3)


12x + 16y = 540 … (4)


Step 2: Subtract equation (4) from equation (3),


(12x – 12x) + (16y - 12y) = 540 – 480


4y = 60


y = 15


Step 3: Substitute y value in (1),


4x – 4(15) – 160 = 0


4x 220 = 0


x = 55


The solution is x = 55, y = 15.

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