Question

1. if ( moverarc{bed} = (15x - 47)^circ ) and ( moverarc{bd} = (9x - 1)^circ ), find ( mangle cbd ).

Explanation:

Step1: Set up full circle equation

The sum of arcs $\overparen{BED}$ and $\overparen{BD}$ is $360^\circ$ (full circle).
$$(15x - 47) + (9x - 1) = 360$$

Step2: Solve for $x$

Combine like terms and isolate $x$.
$$24x - 48 = 360$$
$$24x = 408$$
$$x = \frac{408}{24} = 17$$

Step3: Find $m\overparen{BED}$

Substitute $x=17$ into $\overparen{BED}$ formula.
$$m\overparen{BED} = 15(17) - 47 = 255 - 47 = 208^\circ$$

Step4: Calculate $m\angle CBD$

$\angle CBD$ is an angle formed by a tangent and chord, equal to half its intercepted arc $\overparen{BED}$.
$$m\angle CBD = \frac{1}{2}m\overparen{BED} = \frac{1}{2}(208)$$

Answer:

$104^\circ$