Question

in the diagram below, $overline{ab}$ and $overline{bc}$ are tangent to $odot o$. what is the measure of $angle b$? a. 126° b. 108° c. 54° d. 180°

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle OAB = 90^{\circ}$ and $\angle OCB=90^{\circ}$.

Step2: Find the measure of $\angle AOC$

The measure of an arc is related to the central - angle that subtends it. The sum of the measures of the arcs of a circle is $360^{\circ}$. Given the measure of arc $ADC = 234^{\circ}$, then the measure of arc $AC=360^{\circ}-234^{\circ}=126^{\circ}$. The measure of the central - angle $\angle AOC$ is equal to the measure of arc $AC$, so $\angle AOC = 126^{\circ}$.

Step3: Use the sum of angles in a quadrilateral

In quadrilateral $OABC$, the sum of the interior angles is $(4 - 2)\times180^{\circ}=360^{\circ}$. Let $\angle B=x$. We know that $\angle OAB = 90^{\circ}$, $\angle OCB = 90^{\circ}$, and $\angle AOC = 126^{\circ}$. Then $x+90^{\circ}+90^{\circ}+126^{\circ}=360^{\circ}$.

Step4: Solve for $\angle B$

$x=360^{\circ}-(90^{\circ}+90^{\circ}+126^{\circ})$.
$x = 54^{\circ}$.

Answer:

C. $54^{\circ}$