Exercise 20 Page 492

Let's consider a right triangle ABC, and let's write the tangent ratio of one acute angle.

We are interested in finding the measure of ∠ A for which tan A is equal to 1, greater than 1, and less than 1. Let's study each case separately.

Equal to 1

We begin this case by considering tan A=1 and substituting it into the tangent ratio written above. BC/AC = tan A = 1 ⇒ AC = BC From the above, we conclude that △ ABC is an isosceles triangle.

Since △ ABC is an isosceles right triangle, we have that the measure of both acute angles is 45^(∘).

Greater Than 1

Let's suppose that tan A > 1 and substitute it into the tangent ratio equation. BC/AC = tan A > 1 ⇒ BC > AC From the latter inequality and the Triangle Longer Side Theorem, we conclude that m∠ A > m∠ B.

Now, we will add m ∠ A to both sides of the inequality written above and use the fact that ∠ A and ∠ B are complementary.

That way, we've found the angle measures for which the tangent is greater than 1.

Less Than 1

Finally, let's suppose that tan A < 1 and substitute it into the equation written at the beginning. BC/AC = tan A < 1 ⇒ BC < AC Again, from the latter inequality and the Triangle Longer Side Theorem, we conclude that m∠ A < m∠ B.

As before, we will add m ∠ A to both sides of the inequality above and use the fact that ∠ A and ∠ B are complementary.

From the latter inequality, we can write the following conclusion.