How do you prove that vertical angles are congruent

How do you prove that vertical angles are congruent?

The Correct Answer and Explanation is :

Proving Vertical Angles Are Congruent

Vertical angles are the pairs of opposite angles formed when two lines intersect. To prove that vertical angles are congruent, we can use the Angle Addition Postulate and the properties of linear pairs.

Proof:

1. Identify the Vertical Angles:
   Consider two intersecting lines that form two pairs of opposite (vertical) angles. Let the lines intersect at point ( O ), creating angles ( \angle AOB ), ( \angle BOC ), ( \angle AOC ), and ( \angle BOD ), where ( \angle AOB ) and ( \angle AOC ) are vertical angles, and ( \angle BOC ) and ( \angle BOD ) are vertical angles.
2. Label Linear Pairs:
   Notice that each pair of adjacent angles form a linear pair. A linear pair of angles is formed when two angles share a common side and the other sides form a straight line.

• ( \angle AOB ) and ( \angle BOC ) form a linear pair.
• ( \angle AOC ) and ( \angle BOD ) form a linear pair.

1. Apply the Linear Pair Theorem:
   The Linear Pair Theorem states that if two angles form a linear pair, they are supplementary. This means that the sum of the measures of the two angles is ( 180^\circ ).

• Since ( \angle AOB ) and ( \angle BOC ) are a linear pair, we have:
  [
  \angle AOB + \angle BOC = 180^\circ
  ]
• Similarly, ( \angle AOC ) and ( \angle BOD ) are a linear pair, so:
  [
  \angle AOC + \angle BOD = 180^\circ
  ]

1. Set the Two Equations Equal:
   Both ( \angle AOB + \angle BOC = 180^\circ ) and ( \angle AOC + \angle BOD = 180^\circ ). Since the sum of the angles in each pair is equal to ( 180^\circ ), the measures of ( \angle AOB ) and ( \angle AOC ) must be equal, and likewise, the measures of ( \angle BOC ) and ( \angle BOD ) are equal. Hence, vertical angles ( \angle AOB ) and ( \angle AOC ) are congruent, as are ( \angle BOC ) and ( \angle BOD ).

Conclusion:
Through the use of the Linear Pair Theorem and the Angle Addition Postulate, we can conclude that vertical angles are congruent. This proof relies on the fact that linear pairs are supplementary and that adjacent angles at the intersection point sum to ( 180^\circ ).