Final answer:
The bisectors of a pair of vertically opposite angles are co-linear, which is proven by the fact that vertically opposite angles are equal and thus their bisectors divide them into equal half angles that are aligned.
Step-by-step explanation:
Proof of Bisectors of Vertically Opposite Angles
To prove that the bisectors of a pair of vertically opposite angles are in the same straight line, we can use the properties of vertical angles and the definition of an angle bisector. Let's assume we have two intersecting straight lines that form two pairs of vertically opposite angles. We will label these angles α, β, α', and β' respectively, where α and α' are vertically opposite, as are β and β'.
The bisector of angle α divides it into two angles of equal measure; let's call them α/2. Similarly, the bisector of α' also divides it into two equal angles of α'/2. However, since α and α' are vertically opposite, they are equal (α = α'), which means their half measures are also equal (α/2 = α'/2). Hence, the bisectors lie along the same line, proving that they are indeed co-linear.