Question

in the figure above, $overline{ad}$, $overline{be}$, and $overline{cf}$ intersect at point $o$. if the measure of $angle aob$ is $80^circ$ and $overline{cf}$ bisects $angle bod$, what is the measure of $angle eof$?
(a) $40^circ$
(b) $50^circ$
(c) $60^circ$
(d) $70^circ$
(e) $80^circ$

Explanation:

Step1: Find $\angle BOD$

$\angle AOB + \angle BOD = 180^\circ$ (linear pair)
$\angle BOD = 180^\circ - 80^\circ = 100^\circ$

Step2: Find $\angle BO C$

$\overline{CF}$ bisects $\angle BOD$, so $\angle BOC = \frac{1}{2}\angle BOD$
$\angle BOC = \frac{1}{2} \times 100^\circ = 50^\circ$

Step3: Find $\angle AOF$

$\angle AOF = \angle BOC = 50^\circ$ (vertical angles)

Step4: Find $\angle AOE$

$\angle AOE = \angle AOB = 80^\circ$ (vertical angles)

Step5: Calculate $\angle EOF$

$\angle EOF = 180^\circ - \angle AOF - \angle AOE$
$\angle EOF = 180^\circ - 50^\circ - 80^\circ = 40^\circ$

Answer:

(A) $40^\circ$