Question

Find the measures of each angle in the inscribed quadrilateral.

Answer 1

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°