Exercise 41 Page 237

Let's consider the given information, the statement we want to prove and the diagram. Given:& △ ABC is a right triangle. Prove:& ∠ A and ∠ B are complementary.

According to the Triangle Sum Theorem, the sum of the measures of the three interior angles of a triangle add to 180^(∘). Triangle Sum Theorem m∠ A+m∠ B+m∠ C=180^(∘) We can rearrange the above equation to isolate m∠ C. To do so, we will subtract m∠ A and m∠ B from both sides. This can be done using the Subtraction Property of Equality. Subtraction Property of Equality m∠ C= 180^(∘)-m∠ A-m∠ B By definition of right angle, we know ∠ C measures 90^(∘). Definition of Right Angle m∠ C= 90^(∘) By the Transitive Property of Equality, we can equate the equivalent expressions for m∠ C. Transitive Property of Equality 180^(∘) -m∠ A-m∠ B= 90^(∘) Once again, we can use the Subtraction Property of Equality to rearrange the above equation. In this case, we will subtract 180^(∘) from both sides. Subtraction Property of Equality - m∠ A-m∠ B=- 90^(∘) Finally, and by the Multiplication Property of Equality, we will multiply all the terms in the above equation by - 1. Multiplication Property of Equality m∠ A+m∠ B=90^(∘) By definition of complementary angles, if the measures of two angles add to 90^(∘), they are complementary. Definition of complementary angles ∠ A and ∠ B are complementary We can summarize our proof using a table.