Question

Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

Answer 1

The bisectors of a pair of vertically opposite angles are in the same straight line.

Step-by-step explanation:

Let's consider a pair of vertically opposite angles formed by two intersecting lines. Vertically opposite angles are the angles that are opposite each other when two lines intersect. In the diagram below, angles AOB and COD are vertically opposite angles.

To prove that the bisectors of a pair of vertically opposite angles are in the same straight line, we need to show that the bisectors of angles AOB and COD intersect at a single point.

Using the angle bisector theorem, we know that the bisectors of angles AOB and COD divide the angles into two equal halves. Let's denote the bisectors as AX and CY, where X is on line OA and Y is on line OC.

Since AX and CY are angle bisectors, we have:

angle AOX = angle YOC (equal half-angles)

angle BOX = angle COY (equal half-angles)

Considering triangle AOX, we have:

angle AOX + angle BOX + angle ABO = 180 degrees (sum of angles in a triangle)

Substituting the equal half-angles, we get:

angle YOC + angle COY + angle ABO = 180 degrees

angle YOC + angle COY + angle ABO + angle ACO = 180 degrees (adding angle ACO to both sides)

Using the fact that angle ACO and angle ABO are vertical angles, which are equal, we can simplify the equation to:

angle YOC + angle COY + angle AOC = 180 degrees

This equation shows that the sum of the angles in triangle COY is 180 degrees, which means that the three points X, Y, and O are collinear and lie on the same straight line.