Question

l19 we do lesson
here are 2 intersecting lines that create 2 pairs of vertical angles:

1. what is the relationship between vertical angles? write down a conjecture. label the diagram to make it easier to write your conjecture precisely.

type a response

Explanation:

Step1: Label the diagram

Let the two intersecting lines be \(AB\) and \(CD\) intersecting at point \(O\). Let \(\angle AOC\) and \(\angle BOD\) be one pair of vertical angles, and \(\angle AOD\) and \(\angle BOC\) be the other pair.

Step2: Use angle - sum properties

We know that \(\angle AOC+\angle AOD = 180^{\circ}\) (linear - pair of angles, they form a straight line). Also, \(\angle BOD+\angle AOD=180^{\circ}\) (linear - pair of angles).
From \(\angle AOC+\angle AOD=\angle BOD + \angle AOD\), we can subtract \(\angle AOD\) from both sides of the equation.
Using the subtraction property of equality: \(\angle AOC+\angle AOD-\angle AOD=\angle BOD+\angle AOD - \angle AOD\)
So, \(\angle AOC=\angle BOD\)
Similarly, for the other pair of vertical angles \(\angle AOD\) and \(\angle BOC\), since \(\angle AOD+\angle AOC = 180^{\circ}\) and \(\angle BOC+\angle AOC=180^{\circ}\), by subtracting \(\angle AOC\) from both equations, we get \(\angle AOD=\angle BOC\)

Answer:

Vertical angles are congruent. If two lines intersect, then the pairs of vertical angles formed are equal in measure.