Question

in circle d, $overleftrightarrow{ab}$ is a tangent with point a as the point of tangency and $mangle cab = 105^{circ}$. what is $mwidehat{cea}$? enter your answer in the box.

Explanation:

Step1: Find ∠CAD

First, recall that a tangent is perpendicular to the radius at the point of tangency. So $\angle DAB = 90^\circ$.
Given $m\angle CAB = 105^\circ$, calculate $\angle CAD$:
$\angle CAD = m\angle CAB - \angle DAB = 105^\circ - 90^\circ = 15^\circ$

Step2: Find arc CA

$\angle CAD$ is an inscribed angle intercepting arc $CA$. The measure of an inscribed angle is half the measure of its intercepted arc.
$m\overset{\frown}{CA} = 2 \times \angle CAD = 2 \times 15^\circ = 30^\circ$

Step3: Find arc CEA

The total measure of a circle is $360^\circ$. Subtract the measure of arc $CA$ from $360^\circ$:
$m\overset{\frown}{CEA} = 360^\circ - m\overset{\frown}{CA} = 360^\circ - 30^\circ$

Answer:

$330^\circ$