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Answer 1

$\large\underline{\sf{Solution-}}$

As it is given that,

ST || QR ⇛ ∠STQ = ∠TQR [ Alternate interior angles ]

⇛ ∠STQ = ∠TQR = 27°

Now, we know angle subtended at the centre of a circle by an arc is double the angle subtended on the remaining circumference.

⇛ ∠SOQ = 2∠STQ = 2 × 27 = 54°

Also, ∠TOR = 2∠TQR = 2 × 27 = 54°

As SOT is a straight line.

⇛ ∠SOQ + ∠QOR + ∠ROT = 180°

$\rm \: 54\degree + y + 54\degree = 180\degree$

$\rm \: 108\degree + y = 180\degree$

$\rm \: y = 180\degree - 108\degree$

$\rm\implies \:\boxed{ \rm{ \:y \: = \: 72 \degree \: \: }}$

Also, ∠QOR = 2 ∠QPR

$\rm\implies \:y = 2x$

$\rm\implies \: 2x = 72\degree$

$\rm\implies \:\boxed{ \rm{ \:x\: = \: 36 \degree \: \: }}$

$\rule{190pt}{2pt}$

Additional information :-

1. Angle in same segments are equal.

2. Angle in semi-circle is 90°.

3. Sum of the opposite angles of a cyclic quadrilateral is supplementary.

4. Exterior angle of a cyclic quadrilateral is equals to interior opposite angles.

Answer 2

Answer:

x = 36,y = 72

Step-by-step explanation:

Given ,

ST || QR ⇛ ∠STQ = ∠TQR [ Alternate interior angles ]

⇒ ∠STQ = ∠TQR = 27°

we know,

Angle subtended at the centre of a circle by an arc is double the angle subtended on the remaining circumference.

⇒∠SOQ = 2∠STQ = 2 × 27 = 54°

Also, ∠TOR = 2∠TQR = 2 × 27 = 54°

As SOT is a straight line.

∠SOQ + ∠QOR + ∠ROT = 180°

⇒54 + y + 54 = 180

⇒108 + y = 180

⇒y = 180 - 108

⇒y = 72

Also, ∠QOR = 2 ∠QPR

⇒y = 2x

⇒72 = 2x

⇒72/2 = x

⇒36 = x