Question

find the value of X and y in given figure ​

Answer 1

$\underline{ \underline{ \Large \bf { Answer:} } }$

$\longmapsto \pmb{ \rm \red { Y = 110 }}$

• $\longmapsto \pmb{ \rm \red { X = 35 }}$

$\underline{ \underline{ \Large \bf { Given:} } }$

• Angle Q = Y + 20

• Angle R = Y

• Angle P = 2X

• Angle S = 50

$\underline{ \underline{ \Large \bf { To \: calculate:} } }$

• Value of X and Y.

$\underline{ \underline{ \Large \bf { Calculation:} } }$

Value of Y ::

As we know that,

• The sum pair of opposite angle of a cyclic quadrilateral is 180°.

So,

$\longmapsto \rm {\angle Q + \angle S = {180}^{\circ} }$

$\longmapsto \rm {(Y + 20) + 50= 180}$

$\longmapsto \rm { Y + 70= 180 }$

$\longmapsto \rm { Y = 180 - 70 }$

$\longmapsto \pmb{ \rm \red { Y = 110 }}$

Therefore, value of Y is 110.

Value of X ::

As we know that,

• The sum pair of opposite angle of a cyclic quadrilateral is 180°.

So,

$\longmapsto \rm {\angle P + \angle R = {180}^{\circ} }$

$\longmapsto \rm { 2X+ Y= 180}$

$\longmapsto \rm { 2X + 110 = 180 }$

$\longmapsto \rm { 2X = 180 - 110 }$

$\longmapsto \rm { X = \dfrac{70}{2} }$

$\longmapsto \pmb{ \rm \red { X = 35 }}$

Therefore, value of X is 35.

$\underline{ \underline{ \Large \bf { Verification:} } }$

As we know that,

• Sum of the all angles of any quadrilateral is equivalent to 360°.

So,

LHS:

$\longmapsto \rm {\angle Q + \angle R + \angle P + \angle S }$

$\longmapsto \rm { (Y+20)^{\circ} + Y^{\circ} + 2X^{\circ} + 50^{\circ} }$

Substituting the value of Y and X.

$\longmapsto \rm{ (110+20)^{\circ} + 110^{\circ} + 2(35)^{\circ} + 50^{\circ} }$

$\longmapsto \rm { 130^{\circ} + 110^{\circ} + 70^{\circ} + 50^{\circ} }$

$\longmapsto \pmb{ \rm \red { 360^{\circ} }}$

RHS:

$\longmapsto \pmb{ \rm \red { 360^{\circ} }}$

LHS = RHS

Hence, verified !!

Answer 2

By observation, the vertices of the above quadrilateral lie in a circle. So, it is a cyclic.

Sum of interior opposite angles of a cyclic quadrilateral = 180°

∴ ∠S + ∠Q = 180°

⇒ (Y + 20)° + 50° = 180°

⇒ Y° = 180° - 70°

⇒ Y° = 110°.

Again, ∠P + ∠R = 180°

⇒ (2X°) + (Y°) = 180°

⇒ 2X° + 110° = 180°

⇒ 2X° = 70°

⇒ X° = 35°.

Thus, the values of X° and Y° respectively are 35° and 110°.