Follow the three step process of an indirect proof.
See solution.
Practice makes perfect
We are asked to write an indirect proof of the following statement.
If two angles are complementary,
then neither angle is a right angle.
We can use the three step structure of an indirect proof.
Step 1
In an indirect proof, we assume that the statement is not true. In this case, the statement is a conditional, so it is not true if the condition is true, but the hypothesis is not.
Let's assume that
∠ 1and∠ 2are complementary
and
either∠ 1or∠ 2is a right angle .
Step 2
Using our assumption, we are looking for a contradiction.
Suppose Angle ∠ 1 is a Right Angle
If angle ∠ 1 is a right angle, then m∠ 1=90.
In this case m∠ 1+m∠ 2>90, so the sum of the angle measures is not 90. This means, that ∠ 1 and ∠ 2 are notcomplementary. This contradicts our assumption.
Suppose Angle ∠ 2 is a Right Angle
We can arrive at a similar contradiction if we assume that angle ∠ 2 is a right angle.
Step 3
We showed that angles ∠ 1 and ∠ 2 cannot be complementary if either of these angles is a right angle. This means that our assumption is false. Since our assumption was that the original statement is false, this means that the original statement cannot be false. The original statement must be true.
If two angles are complementary,
then neither angle is a right angle.
