Question

write an inequality describing the range of x for each pair of triangles.
a.
b.
a. x < 27
(simplify your answer.)
b. x <
(simplify your answer.)

Explanation:

Step1: Apply the Hinge Theorem

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
For part b, in \(\triangle EFG\) and \(\triangle ABC\), \(EF = AC\), \(EG=BC\), and \(FG = 34\), \(AB = 41\). Since \(AB>FG\), then \(\angle C>\angle E\).

Step2: Set up the inequality

We know that \(\angle C = 77^{\circ}\), \(\angle E=(3x + 17)^{\circ}\). So, \(77>3x + 17\)

Step3: Solve the inequality

Subtract 17 from both sides: \(77-17>3x+17 - 17\), which gives \(60>3x\).
Divide both sides by 3: \(\frac{60}{3}>x\), so \(20>x\)

Answer:

\(x < 20\)