Question

quadrilateral abcd is inscribed in a circle. m∠a is 64°, m∠b is (5x + 4)°, and m∠c is (9x - 1)°. what is m∠d? a. 64° b. 82° c. 90° d. 98° e. 116°

Explanation:

Step1: Use cyclic quadrilateral property

In a cyclic quadrilateral, opposite angles sum to $180^\circ$. So $m\angle B + m\angle C = 180^\circ$.
$$(6x+4) + (9x-1) = 180$$

Step2: Solve for $x$

Combine like terms and isolate $x$.
$$15x + 3 = 180$$
$$15x = 177$$
$$x = \frac{177}{15} = 11.8$$

Step3: Find $m\angle B$

Substitute $x=11.8$ into $m\angle B$ expression.
$$m\angle B = 6(11.8)+4 = 70.8+4 = 74.8^\circ$$

Step4: Find $m\angle D$

Use cyclic quadrilateral property: $m\angle A + m\angle D = 180^\circ$.
$$64 + m\angle D = 180$$
$$m\angle D = 180 - 64 = 98^\circ$$

Answer:

D. $98^\circ$