Question

in the diagram below, (overline{de}) and (overline{ef}) are tangent to (odot o). what is the measure of (angle def?
a. 117°
b. 58°
c. 63°
d. 126°

Explanation:

Step1: Recall tangent - radius property

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

Step2: Find the measure of $\angle DOF$

The measure of an arc is related to the central - angle. The measure of arc $DF$ is $117^{\circ}$, and the central - angle $\angle DOF$ has the same measure as the arc it subtends, so $\angle DOF = 117^{\circ}$.

Step3: Use the sum of angles in a quadrilateral

In quadrilateral $ODEF$, the sum of the interior angles is $(4 - 2)\times180^{\circ}=360^{\circ}$. Let $\angle DEF=x$. Then, $\angle ODE+\angle DEF+\angle OFE+\angle DOF = 360^{\circ}$. Substituting the known values: $90^{\circ}+x + 90^{\circ}+117^{\circ}=360^{\circ}$.

Step4: Solve for $\angle DEF$

Combining like terms gives $x+297^{\circ}=360^{\circ}$. Subtracting $297^{\circ}$ from both sides, we get $x=\angle DEF = 63^{\circ}$.

Answer:

C. 63°