Answer:
m∠P = 55° , m∠R = 125° , m∠Q = 110° , m∠S = 70°
Step-by-step explanation:
* Lets study the figure
- A circle and inscribed quadrilateral PQRS
- The four vertices of the quadrilateral lie on the circumference
of the circle
∴ PQRS is a cyclic quadrilateral
* Lets revise the properties of the cyclic quadrilateral
- Each two opposite angles are supplementary, that means
the sum of each two opposite angles = 180°
- The measure of the exterior angle = the measure of the
opposite interior angle
* Lets solve the problem
∵ PQRS is a cyclic quadrilateral
∴ m∠P + m∠R = 180° ⇒ opposite angles
∵ m∠P = 5y + 30
∵ m∠R = 15y + 50
∴ 5y + 30 + 15y + 50 = 180 ⇒ add the like terms
∴ 20y + 80 = 180 ⇒ subtract 80 from both sides
∴ 20y = 100 ⇒ divide both sides by 20
∴ y = 5
* Now lets substitute the value of y in each angle to find
their measures
∵ m∠P = 5y + 30
∴ m∠P = 5(5) + 30 = 25 + 30 = 55°
∵ m∠R = 15y + 50
∴ m∠R = 15(5) + 50 = 75 + 50 = 125°
∵ m∠Q = y² + 85
∴ m∠Q = 5² + 85 = 25 + 85 = 110°
∵ ∠S is opposite to ∠Q
∴ m∠S + m∠Q = 180° ⇒ opposite angles in the cyclic quadrilateral
∴ m∠S + 110 = 180 ⇒ subtract 110 from the both sides
∴ m∠S = 70°
* The measures of the angles are
m∠P = 55° , m∠R = 125° , m∠Q = 110° , m∠S = 70°
