McGraw Hill Integrated II, 2012
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McGraw Hill Integrated II, 2012 View details
1. Ratios and Proportions
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Exercise 59 Page 549

Consider the Converse of the Hinge Theorem. Remember that since two sides have the same length, they are congruent.

- 4/7 < x < 136/7

Practice makes perfect

We will start by looking closer at the given triangles. Note that if two sides have the same length, they are congruent. Let's mark it on the diagram.

For the given triangles, we will find the range of possible values for x. In order to do that, we will use the following theorem.

Converse of the Hinge Theorem

If two sides of a triangle are congruent to two sides of another triangle, and the third side in one of the triangles is longer than the third side in the other triangle, then the included angle in first triangle is larger than the included angle of the second triangle.

Applying the theorem, we can order the included angles. 12 &> 8 &⇓ 140^(∘) &> (7x+4)^(∘) Let's solve the inequality for x.
140>7x+4
136>7x
136/7>x
x<136/7
As a result, all x-values less than 1367 will work with the known triangle measurements. Also, notice that (7x+4)^(∘) must be greater than 0^(∘).
7x+4>0
7x>- 4
x>- 4/7
Therefore, - 47 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>- 47 as - 47