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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

6. Inequalities in Two Triangles

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Exercise 18 Page 459

Consider the Converse of the Hinge Theorem.

- 4.5< x< 33

Practice makes perfect

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. 15 > 11 ⇒ 75 ^(∘) > (2x+9) ^(∘) Let's solve the inequality for x.

75 > 2x+9

LHS-9>RHS-9

66 > 2x

.LHS /2.>.RHS /2.

33 > x

Rearrange inequality

x < 33

All x-values less than 33 will work with the known triangle measurements. Additionally, (2x+9)^(∘) must be greater than 0^(∘).

2x+9>0

LHS-9>RHS-9

2x >- 9

.LHS /2.>.RHS /2.

x > - 4.5

Therefore, - 4.5 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>- 4.5 as - 4.5 < x to make it a bit easier. Inequality I:&& & x< 33 Inequality II:&& - 4.5< & x Combined:&& - 4.5<& x< 33

Subchapter links

Exercises p.457-462