Question

lines cd and de are tangent to circle a, as shown below: if arc ce is 105°, what is the measure of ∠cde? 52.5° 62.5° 75° 255°

Explanation:

Step1: Recall tangent - radius property

The radius is perpendicular to the tangent at the point of tangency. So, $\angle ACD = 90^{\circ}$ and $\angle AED=90^{\circ}$.

Step2: Find the measure of $\angle CAE$

The measure of an arc is equal to the measure of the central angle that subtends it. So, the central - angle $\angle CAE$ corresponding to arc $\overset{\frown}{CE}$ is $105^{\circ}$.

Step3: Use the sum of angles in a quadrilateral

In quadrilateral $ACDE$, the sum of the interior angles is $360^{\circ}$. Let $\angle CDE=x$. Then, $\angle ACD+\angle CAE+\angle AED+\angle CDE = 360^{\circ}$. Substituting the known values: $90^{\circ}+105^{\circ}+90^{\circ}+x = 360^{\circ}$.

Step4: Solve for $\angle CDE$

Combining like - terms: $285^{\circ}+x = 360^{\circ}$. Subtracting $285^{\circ}$ from both sides gives $x=360^{\circ}-285^{\circ}=75^{\circ}$.

Answer:

C. $75^{\circ}$