Proving Congruency Between Two Supplementary Angles: Theorem and Proof

If two angles are supplementary to congruent angles, then

we need to prove that the two angles are congruent

we need to prove that the two angles are congruent.
To begin, let’s define some terms:
Supplementary angles: Two angles are supplementary if the sum of their measures is equal to 180 degrees.
Congruent angles: Two angles are congruent if their measures are equal.

Given: Two angles, let’s call them angle A and angle B, are supplementary to congruent angles.

We need to prove: Angle A and angle B are congruent.

Proof:
1. Let’s assume that the congruent angles are angle C and angle D.
2. According to the given statement, angle A and angle B are both supplementary to angle C.
3. By the definition of supplementary angles, we have: angle A + angle C = 180 degrees.
4. Similarly, angle B and angle C are supplementary, so we have: angle B + angle C = 180 degrees.
5. Combining equations (3) and (4), we get: angle A + angle C = angle B + angle C.
6. By subtracting angle C from both sides of the equation (5), we have: angle A = angle B.
7. Therefore, angle A and angle B are congruent.

Hence, we have proved that if two angles are supplementary to congruent angles, then the two angles are congruent.

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