Question

what is the measure of $\angle bvd?$
a. $105^\circ$
b. $95^\circ$
c. $75^\circ$
d. $115^\circ$

Explanation:

Step1: Identify angle relationship

∠AVC and ∠BVD are adjacent supplementary angles? No, wait, ∠AVC (105°) and ∠BVD: actually, ∠AVC and ∠BVD are same - side? Wait, no, lines AB and CD intersect at V. So ∠AVC and ∠BVD: wait, ∠AVC and ∠BVD are supplementary? Wait, no, ∠AVC and ∠BVD: let's see, ∠AVC + ∠BVD = 180°? Wait, no, ∠AVC and ∠BVD: actually, ∠AVC and ∠BVD are adjacent to a straight line. Wait, the angle between AV and CV is 105°, and we need to find ∠BVD. Since AB and CD are straight lines, ∠AVC and ∠BVD are supplementary? Wait, no, ∠AVC and ∠BVD: let's think again. The sum of adjacent angles on a straight line is 180°. So ∠AVC + ∠BVD = 180°? Wait, no, ∠AVC and ∠BVD: actually, ∠AVC and ∠BVD are vertical angles? No, vertical angles are equal. Wait, maybe I made a mistake. Let's look at the diagram. AB is a straight line, CD is a straight line, intersecting at V. So ∠AVC and ∠BVD: ∠AVC is 105°, and ∠BVD is adjacent to it on the straight line? Wait, no, ∠AVC and ∠BVD: the angle between AV and CV is 105°, so the angle between BV and DV should be 180° - 105° = 75°? Wait, no, wait. Wait, ∠AVC and ∠BVD: are they supplementary? Let's see, if we consider the straight line AB, then the angles on one side of AB should sum to 180°. Wait, ∠AVC is 105°, so the angle adjacent to it (∠BVC) would be 180° - 105° = 75°, but no, wait, ∠BVD: maybe I confused the angles. Wait, let's label the intersection. Points A and B are on a straight line, points C and D are on a straight line, intersecting at V. So ∠AVC and ∠BVD: ∠AVC is 105°, so ∠BVD is equal to 180° - 105°? Wait, no, ∠AVC and ∠BVD: are they supplementary? Let's calculate 180 - 105 = 75. So ∠BVD should be 75°.

Step2: Calculate the angle

We know that the sum of two adjacent angles on a straight line is 180°. So if one angle (∠AVC) is 105°, then the other angle (∠BVD) is $180^{\circ}- 105^{\circ}=75^{\circ}$.

Answer:

C. 75°