Question

Prove the following lemma: "Congruent angles that form a linear pair are right angles." (Suggestion: Write this as a conditional statement, naming your angles, so you are clear what you are to assume.)

A) If two congruent angles form a linear pair, then they are right angles.
B) If two right angles form a linear pair, then they are congruent.
C) If two angles form a linear pair, then they are congruent.
D) If two congruent angles form a linear pair, then they are obtuse angles.

Answer 1

Two congruent angles that form a linear pair are right angles because their measures add up to 180 degrees, resulting in each angle having a measure of 90 degrees.

Step-by-step explanation:

The statement to prove is: If two congruent angles form a linear pair, then they are right angles.

Proof:

1. Let's assume that we have two congruent angles, angle A and angle B, that form a linear pair. This means that angle A and angle B are adjacent angles and their non-common sides form a straight line.
2. Since angle A and angle B are congruent, their measures must be equal. Let's say their measure is x.
3. Since angle A and angle B form a straight line, the sum of their measures must be 180 degrees. Therefore, we can write the equation x + x = 180.
4. Simplifying the equation, we get 2x = 180. Dividing both sides by 2, we find that x = 90.
5. Since angle A and angle B have a measure of 90 degrees, they are right angles.
6. Therefore, we have proven that if two congruent angles form a linear pair, then they are right angles.