Final answer:
To prove that ∠3 and ∠4 are complementary angles, we need to show that their sum is 90 degrees. By using the given information that ∠1 and ∠2 are complementary, we can substitute their values into the equation and simplify it to show that ∠3 + ∠4 equals 90 degrees.
Step-by-step explanation:
Complementary angles are two angles whose sum is 90 degrees. In this case, we are given that ∠1 and ∠2 are complementary angles. To prove that ∠3 and ∠4 are also complementary angles, we need to show that their sum is 90 degrees.
Since ∠1 and ∠2 are complementary, we know that ∠1 + ∠2 = 90 degrees. We can represent ∠3 and ∠4 as follows: ∠3 = 90 - ∠1 and ∠4 = 90 - ∠2. Substituting these values, we get: ∠3 + ∠4 = (90 - ∠1) + (90 - ∠2).
Using the distributive property of addition, we can simplify the equation: ∠3 + ∠4 = 90 - ∠1 + 90 - ∠2. Rearranging and combining like terms, we have ∠3 + ∠4 = 180 - (∠1 + ∠2).
Since ∠1 + ∠2 = 90 degrees from the given information, we can substitute this into the equation: ∠3 + ∠4 = 180 - (90). Simplifying further, we get ∠3 + ∠4 = 180 - 90 = 90 degrees.
This shows that the sum of ∠3 and ∠4 is indeed 90 degrees, which means that ∠3 and ∠4 are complementary angles.