Introduction:
Vertically opposite angles are formed when two lines intersect. They are opposite each other at the point of intersection, and their angles are equal. The bisectors of vertically opposite angles are lines that divide these angles into two equal parts. In this proof, we will show that these bisectors are in the same straight line.

Proof:
Let's consider two lines, AB and CD, that intersect at point O. This intersection creates four angles: ∠AOC, ∠AOD, ∠BOC, and ∠BOD.

Claim 1: ∠AOC = ∠BOD and ∠AOD = ∠BOC
Proof of Claim 1: Since AB and CD intersect at point O, the opposite angles formed are equal. Therefore, ∠AOC = ∠BOD and ∠AOD = ∠BOC.

Claim 2: The bisectors of ∠AOC and ∠AOD are in the same straight line.
Proof of Claim 2: Let the bisector of ∠AOC be line OE, and the bisector of ∠AOD be line OF. We need to prove that these bisectors are in the same straight line, i.e., line OE is coincident with line OF.

Claim 3: ∠EOC = ∠FOC
Proof of Claim 3: Since OE is the bisector of ∠AOC, it divides ∠AOC into two equal parts. Therefore, ∠EOC = ∠AOC/2. Similarly, OF is the bisector of ∠AOD, so ∠FOC = ∠AOD/2.

Claim 4: ∠AOC = ∠AOD
Proof of Claim 4: From Claim 1, we know that ∠AOC = ∠BOD and ∠AOD = ∠BOC. By transitive property, we can conclude that ∠AOC = ∠AOD.

Claim 5: ∠EOC = ∠FOC
Proof of Claim 5: From Claim 3 and Claim 4, we can see that ∠EOC = ∠AOC/2 = ∠AOD/2 = ∠FOC.

Conclusion:
Based on Claim 5, we have shown that the bisectors of ∠AOC and ∠AOD intersect at point O, and therefore, they are in the same straight line. Hence, the bisectors of a pair of vertically opposite angles lie in the same straight line.