Question

If two angles are supplementary to congruent angles (or to the same angle), then they are congruent. Proof: a) Angle addition postulate b) Angle bisector theorem c) Corresponding angles theorem d) Vertical angles theorem

Answer 1

When two angles are supplementary to congruent angles, they are congruent because they would both be equal to 180 degrees minus the measure of the congruent angles they are supplementary to.

Step-by-step explanation:

Understanding Supplementary Angles and Congruence

If two angles are supplementary to congruent angles, or to the same angle, they add up to 180 degrees each, by the definition of supplementary angles. The Angle Addition Postulate states that if angle A is equal to angle B and both are supplementary to angles C and D respectively, then angle C and D would have to be congruent because they both would equal 180 degrees minus the measurement of angle A (or B). Mathematically, if angle A equals angle B, then angle C equals 180 degrees minus angle A and angle D equals 180 degrees minus angle B. Since angles A and B are congruent, their measures are equal, so the deductions for angles C and D would also be equal, making them congruent.

Answer 2

To prove the congruence of two angles that are supplementary to congruent angles, we can use the Angle Addition Postulate, Angle Bisector Theorem, Corresponding Angles Theorem, and Vertical Angles Theorem step-by-step.

Step-by-step explanation:

To prove that if two angles are supplementary to congruent angles or to the same angle, then they are congruent, we can make use of the Angle Addition Postulate, the Angle Bisector Theorem, the Corresponding Angles Theorem, and the Vertical Angles Theorem. Let's go step-by-step:

1. According to the Angle Addition Postulate, if two angles are supplementary to the same angle, then their sum is equal to that angle. Thus, if angle A and angle B are supplementary to angle C, we have: angle A + angle B = angle C.
2. Now, let's use the Angle Bisector Theorem. If angle C is congruent to angle D, and angle A and angle B are supplementary to angle C, then angle A is congruent to angle B by the Angle Bisector Theorem.
3. According to the Corresponding Angles Theorem, if angle A and angle B are congruent, and angle A and angle E are corresponding angles, then angle B and angle F are also corresponding angles. Thus, angle B is congruent to angle F.
4. Finally, according to the Vertical Angles Theorem, if angle B and angle F are congruent, then angle A and angle C are also congruent, as they are vertical angles.

Therefore, we have proven that if two angles are supplementary to congruent angles or to the same angle, then they are congruent.