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Rule

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then they are congruent.
Three labeled angles

Based on the diagram above, the following relation holds true.

If and are supplementary and and are supplementary, then

Proof

Consider two pairs of supplementary angles: and and and
Three labeled angles
By the definition of supplementary angles, the sum of the angle measures in each of these pairs is
The right-hand sides of the equations are the same. Therefore, by the Transitive Property of Equality the left-hand sides must also be equal.
Finally, by using the Subtraction Property of Equality can be subtracted from each side of the obtained equation.
The measures of and are the same, which indicates that these angles are congruent.