We will write a proof for the Complement Theorem. First, let's remember what the Complement Theorem states.
Complement Theorem
If the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles.
Let's make a general diagram that illustrates this situation.
Here, the non-common sides of ∠ 1 and ∠ 2 form a right angle. In other words, the measure of the ∠ ABC is 90^(∘). Then, by using the Angle Addition Postulate we obtain the following relation.
m∠ 1 + m∠ 2 &= m∠ ABC
&⇓
m∠ 1 + m∠ 2 &= 90
The right-hand side of the equation tells us that ∠ 1 and ∠ 2 are complementary angles, and this is what we wanted to prove.
Alternative Solution
Alternative Solution
We can also write this proof using a two-column proof table. First, using the diagram above, let's write the given information and what we want to prove.
Given:& ∠ ABC is a right angle
Prove:& ∠ 1 and ∠ 2 are complementary
Now we can proceed with the proof.
