A Detailed Proof about Congruent Angles

In this post, I want to explain in details how to prove about congruent angles. In the diagram below, \angle 1 and \angle 2 are complementary angles. Also, \angle 3 and \angle 4 are complementary angles. Now, if \angle 1 is congruent to \angle 3, prove that \angle 2 is congruent to angle \angle 4.

congruent angles

Proof

From the problem above, we have the following facts (given).

\angle 1 and \angle 2 are complementary
\angle 3 and \angle 4 are complementary
\angle 1 \cong \angle 3

We want to prove that \angle 2 \cong \angle 4.

1. Statement: m \angle 1 = m \angle 3
Reason: Definition of Congruent Angles

Explanation: The symbol \cong means congruent which means that we are not really talking about the actual measure, but about the size of the angles. This is the same as saying that my shoe size is the same as yours, without actually mentioning the size of the shoes.  The statement m \angle 1 = m \angle 3 makes it explicit that we know that the measure of angle 1 is the same as the measure of angle 2 (say in degrees). The definition of congruent angles states that congruent angles have the same measure.

2. Statement

2a: m \angle 1 + m \angle 2 = 90
2b: m \angle 3 + m \angle 4 = 90

Reason: Definition of Complementary Angles

Explanation: Angles 1 and 2 are complementary angles, therefore the sum of their measures equals 90 degrees. This is also the same as angles 3 and 4.

3.  Statement: m \angle 1 + m \angle 2 = m \angle 3 + m \angle 4
Reason: Substitution

Explanation: Both pairs of angles add up to 90 degrees, so their sums are equal. Therefore, we can just substitute the statement in 2b to the right hand side of statement 2a.

4. Statement: m \angle 1 + m \angle 2 = m \angle 1 + m \angle 4
Reason: Substitution

Explanation: Notice that the statement 4 is exactly the same as statement 3 except that m \angle 3 on the right hand side of the equation was changed to m \angle 1. This is because m \angle 1 = m \angle 3 (statement 1).

5. Statement: m \angle 2 = m \angle 4

Reason: Subtraction Property of Equality

Notes: We subtracted m \angle 1 from both sides leaving m \angle 2 on the left hand side and m \angle 4 on the right hand side.

6. Statement: \angle 2 \cong \angle 4.
Reason: Definition of Congruent Angles

Notes: Again, congruent angles have equal measures and the other way around. Angles with equal measures are congruent.

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