Final answer:
In a circle with a bisecting segment MA, we can prove that angle A is congruent to angle B, angle A is congruent to angle C, angle B is congruent to angle MAC, and angle B is congruent to angle C. Therefore, the correct option is d).
Step-by-step explanation:
In the given scenario, we have a circle M, and segment MA bisects segment BC. We need to prove certain congruencies between angles.
a. To prove Angle A is congruent to angle B, we can use the fact that segment MA bisects segment BC. This means that segment BA is equal in length to segment AC.
As a result, angle B is congruent to angle C (by the Isosceles Triangle Theorem).
Since angle A and angle C are opposite angles in a parallelogram (formed by extending segment MA and segment BC), angle A is congruent to angle B.
b. To prove Angle A is congruent to angle C, we can use the same reasoning as in part a.
Since angle B and angle C are congruent (proved in part a), and angle A and angle B are congruent (proved by the transitive property of congruence), angle A is also congruent to angle C.
c. To prove Angle B is congruent to angle MAC, we can use the Angle Bisector Theorem.
Since segment MA bisects segment BC, it divides angle MAC into two congruent angles. Therefore, angle B is congruent to angle MAC.
d. To prove Angle B is congruent to Angle C, we can again use the transitive property of congruence.
Angle B is congruent to angle MAC (proved in part c), and angle MAC is congruent to angle C (proved in part a).
Therefore, angle B is congruent to angle C. Therefore, the correct option is d).