Question

semester b
in quadrilateral pqrs, ∠pqr measures (7x - 2)°. angle psr measures (5x + 14)°
what are the measures of angles pqr and psr?
options: m∠pqr = 90° and m∠psr = 90°; m∠pqr = 96° and m∠psr = 84°; m∠pqr = 84° and m∠psr = 96°; m∠pqr = 54° and m∠psr = 54°

Explanation:

Step1: Identify cyclic quadrilateral property

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary, i.e., their sum is \(180^\circ\). So, \(\angle PQR + \angle PSR = 180^\circ\). Substitute the given expressions: \((7x - 2) + (5x + 14) = 180\).

Step2: Solve for x

Combine like terms: \(12x + 12 = 180\). Subtract 12 from both sides: \(12x = 168\). Divide by 12: \(x = 14\).

Step3: Calculate \(\angle PQR\)

Substitute \(x = 14\) into \((7x - 2)^\circ\): \(7(14) - 2 = 98 - 2 = 96^\circ\).

Step4: Calculate \(\angle PSR\)

Substitute \(x = 14\) into \((5x + 14)^\circ\): \(5(14) + 14 = 70 + 14 = 84^\circ\). Check: \(96 + 84 = 180\), which satisfies the cyclic quadrilateral property.

Answer:

\(m\angle PQR = 96^\circ\) and \(m\angle PSR = 84^\circ\) (the second option)