Question

applying the converse of the hinge theorem
given \\(\overline{ac} \cong \overline{fe}\\) and \\(\overline{cb} \cong \overline{ed}\\) which statement is correct?
\\(\circ\\) angle a is larger than angle b.
\\(\circ\\) angle c is congruent to angle e.
\\(\circ\\) angle c is smaller than angle e.
\\(\circ\\) angle c is larger than angle e.
(image of two triangles: triangle abc with ac = 15 in, and triangle fde with fe = 11 in, and marked congruent sides)

Explanation:

Step1: Identify given congruent sides

$\overline{AC} \cong \overline{FE}$ and $\overline{CB} \cong \overline{ED}$ (marked sides).

Step2: Compare opposite sides of angles

Side opposite $\angle C$ is $AB = 15$ in; side opposite $\angle E$ is $FD = 11$ in. $15 > 11$.

Step3: Apply Converse of Hinge Theorem

If two sides of one triangle ≅ two sides of another, the larger included angle is opposite the longer third side. So $\angle C > \angle E$.

Answer:

D. Angle C is larger than angle E.