In a cyclic quadrilateral ABCD, if = (2x-1)^ , =(y+5)^ , = (2y+15… - Vidyadip

MCQ

In a cyclic quadrilateral $\text{ABCD},$ if $\angle\text{A} = (2\text{x}-1)^\circ,\angle\text{B}=(\text{y}+5)^\circ,\angle\text{C} = (2\text{y}+15)^\circ$ and $\angle\text{D}= (4\text{x}-7)^\circ$ then the value of $\angle\text{C}$ is :

A
$125^\circ$
B
$65^\circ$
✓
$115^\circ$
D
$55^\circ$

✓

Answer

Correct option: C.

$115^\circ$

Since the sum of the opposite angles of a cyclic quadrilateral is $180^\circ$
$\therefore\angle\text{A}+\angle\text{C} = 180^\circ$
$\Rightarrow 2x - 1 + 2y + 15 = 180^\circ$
$\Rightarrow x + y = 83^\circ ... (i)$
And $\angle\text{B}+\angle\text{D} = 180^\circ$
$\Rightarrow y + 5 + 4x - 7 = 180^\circ$
$\Rightarrow 4x + y = 182^\circ ... (ii)$
Subtracting eq. $(ii)$ from eq. $(i),$
we get $ -3x = -99^\circ$
$\Rightarrow x = 33^\circ$
Putting the value of $x$ in eq. $(i),$
we get $33^\circ + y = 83^\circ$
$\Rightarrow y = 50^\circ$
$\therefore\angle\text{C} = (2\text{y}+15)^\circ$
$=(2\times50+15)^\circ = 115^\circ$

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