Exercise 19 Page 459

For the given triangles, we will find the range of possible values for x.

In order to do that, we will use the Converse of the Hinge Theorem to compare the included angles.

Applying the theorem, we can write an inequality for the included angles. 54 > 28 ⇒ 41 ^(∘) > (x+20) ^(∘) Let's solve the inequality for x.

All x-values less than 21 will work with the known triangle measurements. besides that (x+20)^(∘) must be greater than 0^(∘). x+20>0 ⇒ x > - 20 Therefore, - 20 is the lower limit of x. We can determine the range of possible values of x by combining the two inequalities. We will rewrite x>- 20 as - 20 < x to make it a bit easier. Inequality I:&& & x< 21 Inequality II:&& - 20 < & x Combined:&& - 20 <& x< 21