Question

in circle a, the measure of ∠bad is 148°. if m\\(widehat{bc}\\) is 102°, what is m\\(widehat{cd}\\)?

Explanation:

Step1: Recall arc - angle relationship

The measure of a central angle is equal to the measure of its intercepted arc. So, the measure of arc $\widehat{BD}$ is equal to the measure of $\angle BAD$, which is $148^{\circ}$.

Step2: Use the property of arc addition

The sum of the measures of arcs in a circle is $360^{\circ}$, and for the arcs $\widehat{BC}$, $\widehat{CD}$ and $\widehat{BD}$, we know that $\widehat{BC}+\widehat{CD}=\widehat{BD}$. Let $m\widehat{CD}=x$. We are given that $m\widehat{BC} = 102^{\circ}$ and $m\widehat{BD}=148^{\circ}$. Then, using the equation $m\widehat{BC}+m\widehat{CD}=m\widehat{BD}$, we substitute the known values: $102 + x=148$.

Step3: Solve for $x$

Subtract 102 from both sides of the equation $102 + x=148$. We get $x=148 - 102$.
$x = 46^{\circ}$

Answer:

$46^{\circ}$