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

**Proof**

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

and are complementary

and are complementary

We want to prove that .

**1. Statement**:

**Reason: **Definition of Congruent Angles

* Explanation:* The symbol 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 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: **

**2b: **

**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:**

**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:**

**Reason: **Substitution

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

**5. Statement:**

**Reason:** Subtraction Property of Equality

* Notes*: We subtracted from both sides leaving on the left hand side and on the right hand side.

6. **Statement:** .

**Reason:** Definition of Congruent Angles

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