Answer:
The values of x and y are x = 95 and y = 25
Explanation:
If a quadrilateral is inscribed in a circle which means all its vertices lie on the circumference of the circle, then it is called a cyclic quadrilateral.
The properties of the cyclic quadrilateral
- Every two opposite angles are supplementary (the sum of their measures is 180°)
- The measure of an exterior angle at one vertex equals the measure of the interior opposite angle to this vertex
In the given figure
∵ The four vertices of the quadrilateral QRTU lie on the circle
∴ QRTU is a cyclic quadrilateral
∵ ∠ PQU is an exterior angle of quadrilateral QRTU at Q
∵ ∠RTU is the opposite interior angle to it
→ By using the 2nd property above
∴ m∠PQU = m∠RTU
∵ m∠PQU = 95°
∵ m∠RTU = x°
∴ x = 95
∵ ∠QRT and ∠QUT are opposite angles in the cyclic quadrilateral
→ By using the 1st property above
∴ m∠QRT + m∠QUT = 180° ⇒ supplementary angles
∵ m∠QRT = 110°
∴ 110 + m∠QUT = 180°
→ Subtract 110 from both sides
∴ m∠QUT = 70°
In ΔSQU
∵ ∠ PQU is an exterior angle to the Δ T vertex Q
→ That means its measure equal the sum of the measures of the
two opposite interior angles QSU and QUS
∴ m∠PQU = m∠QSU + m∠QUS
∵ m∠PQU = 95°
∵ m∠QSU = y°
∵ m∠QUS = 70°
∴ 95 = y + 70
→ Subtract 70 from both sides
∴ 25 = y
∴ y = 25