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.
