We will prove one of the Right Angles Theorem which states the following.
If two congruent angles form a linear pair, then they are right angles.
To prove it, we will use the diagram below.
For our purposes, it is enough to prove the theorem using only ∠1 and ∠2.
Given:& ∠1 ≅ ∠2 and they form a linear pair
Prove:& ∠1 and ∠2 are right angles
With this information and the Supplement Theorem we will be able to prove this theorem by using the two-column proof table.
Statements
Reasons
1.
∠1 ≅ ∠2 and they form a linear pair
1.
Given
2.
m∠1 = m∠2
2.
Definition of Congruent Angles
3.
m∠1 + m∠2 = 180
3.
Supplement Theorem
4.
m∠1 + m∠1 = 180
4.
Substitution Method
5.
m∠1 = 90
5.
Simplify
6.
m∠2 = 90
6.
Transitive Property of Equality
Alternative Solution
Another Way
Now we will prove this theorem without using two-column proof table. Let's consider the figure above. We know that ∠1 and ∠2 are congruent and they form a linear pair. By the definition of the congruent angles, we know that the measures of the angles are equal. m∠1 = m∠2
Besides by the Supplement Theorem, the sum of the measures of the angles ∠1 and ∠2 are 180^(∘). We will substitute m∠1 for m∠2 to find m∠1.
