The Complementary Angle Theorem | Explained and Applied

Name the theorem: If two angles are complements of the same angle (or of congruent angles) then the two angles are congruent.

The theorem you are referring to is called the Complementary Angle Theorem

The theorem you are referring to is called the Complementary Angle Theorem.

The Complementary Angle Theorem states that if two angles are complements of the same angle, or if they are complements of congruent angles, then the two angles themselves are congruent.

To understand this theorem, let’s begin with the definition of complementary angles. Two angles are said to be complementary if their measures add up to 90 degrees. In other words, the sum of the measures of the two complementary angles is equal to 90 degrees.

Now, if we have two angles, let’s call them angle A and angle B, and if both of them are complementary to the same angle (let’s say angle C), then we can say that angle A + angle C = 90° and angle B + angle C = 90°. Since both equations equal 90 degrees and we know that angle A + angle C = angle B + angle C, we can subtract angle C from both sides of each equation and conclude that angle A = angle B. Therefore, angle A and angle B are congruent.

Similarly, if angle A and angle B are both complementary to congruent angles (which means their measures are the same), then we can use the same reasoning to conclude that angle A = angle B.

In summary, the Complementary Angle Theorem states that if two angles are complements of the same angle or of congruent angles, then the two angles themselves are congruent.

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