The Vertical Angles Theorem: Geometry Rule Explained

Vertical Angles Theorem

If two angles form a pair of vertical angles, then they are congruent

The Vertical Angles Theorem is a geometry rule that states that when two straight lines intersect, they form two pairs of opposite angles that are equal in measure. These opposite angles, also known as vertical angles, are formed when two lines intersect and are located directly across from each other.

The theorem states that if two lines intersect at a point, then the vertical angles formed are congruent or have the same measure. More precisely, if angle A and angle C are vertical angles, then angle A is congruent to angle C.

To prove the theorem, one can use the fact that opposite angles are created when two lines intersect and that the sum of all angles around a point is 360 degrees. Suppose that line AB intersects line CD at point O. Then angles AOC and BOD are a pair of opposite or vertical angles. Similarly, angles AOD and BOC are also vertical angles. Since the sum of the angles at point O is 360 degrees, we can write:

angle AOC + angle COD + angle BOD + angle AOD = 360 degrees

Since AOC and BOD are vertical angles, they have the same measure or are congruent, as do angles AOD and BOC. Therefore, we can rewrite the equation above as:

2(angle AOC) + 2(angle BOC) = 360 degrees

Simplifying, we get:

angle AOC + angle BOC = 180 degrees

Therefore, angle AOC and angle COD form a linear pair, and angle BOD and angle AOD form another linear pair. Since linear pairs are supplementary, we have:

angle AOC + angle COD = 180 degrees

angle BOD + angle AOD = 180 degrees

Hence, we have that angle AOC is congruent to angle COD and that angle BOD is congruent to angle AOD, which proves the Vertical Angles Theorem.

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