An important question from linear equations in two variables as it was already asked in various examinations in which we have been asked to find the angles of a cyclic quadrilateral ABCD if it is given that in a cyclic quadrilateral ABCD, ∠A=(2x+4),∠B=(y+3),∠C=(2y+10),∠D=(4x−5).

Kindly solve the above problem by using the properties of linear equations in two variables

RS Aggarwal, Class 10, chapter 3E, question no 53

Let ABCD be a cyclic quadrilateral.

∠A=2x+4,∠B=y+3,∠C=2y+10,∠D=4x−5

In cyclic quadrilateral the sum of the opposite angles in 180°. Therefore,

∠A+∠C=180°

⇒2x+4+2y+10=180°

⇒2x+2y=166°

⇒x+y=83°→1

∠B+∠D=180°

⇒y+3+4x−5=180°

⇒4x+y=182°→2

Solving 1 and 2, we get

4x+y−x−y=182°−83°

⇒3x=99°

⇒x=33°

& 33°+y=83°

⇒y=83°−33°

=50°

∴∠A=2×33°+4=70°,

∠B=50°+3=53°

∠C=2×50°+10=110°,

∠D=4×33°−5=127°