Question

Quadrilateral MRPN is cyclic, ∠R = (5x – 13)° and ∠N = (4x + 4)°. Find measures of ∠R and ∠N.

Answer 1

HINT:- Before solving this question, we must know about Cyclic Quadrilaterals.
CYCLIC QUADRILATERAL: If all the four vertices of a quadrilateral lie on the circumference of the circle then the quadrilateral is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined they form vertices of a cyclic quadrilateral.

                

Complete step-by-step solution -
As we can see that angle R and angle N are opposite to each other and we know that in cyclic quadrilaterals, the sum of the pair of the opposite angles is supplementary, i.e. 180°.
So, ∠R + ∠N = 180°
(5x – 13)° + (4x + 4)° = 180°
5x – 13 + 4x + 4 = 180°
5x + 4x – 13 + 4 = 180°
9x – 9 = 180°
9x = (180 + 9)°
\[x=\dfrac{189}{9}\]
x = 21
∵ ∠R = (5x – 13)° = \[\left( 5\times 2113 \right){}^\circ \] = (105 – 13)° = 92°
And ∠N = (4x + 4)° = \[\left( 4\times 21+4 \right){}^\circ \] = (84 + 4)° = 88°
Hence, ∠R = 92° and ∠N = 88°

NOTE:- Here are some of the properties of a Cyclic Quadrilateral:-
In a cyclic quadrilateral, the sum of the pair of opposite angles is supplementary, i.e. 180°.
The four vertices of a cyclic quadrilateral lie on the circumference of the circle.
In a cyclic quadrilateral, the four perpendicular bisectors of the given four sides meet at the center O.