Question

in the triangles, \\( \overline { a c } \cong \overline { f e } \\) and \\( \overline { b c } \cong \overline { d e } \\).
which statement correctly compares \\( \overline { a b } \\) and \\( \overline { f d } \\)?
\\( \overline { a b } \\) and \\( \overline { f d } \\) are the same length.
\\( \overline { a b } \\) is longer than \\( \overline { f d } \\).
\\( \overline { a b } \\) is shorter than \\( \overline { f d } \\).
\\( \overline { a b } \\) is shorter than or the same length as \\( \overline { f d } \\).

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 side opposite the larger included angle is longer.
In $\triangle ABC$ and $\triangle FDE$, we have $\overline{AC}\cong\overline{FE}$ and $\overline{BC}\cong\overline{DE}$. The included angle for $\overline{AB}$ in $\triangle ABC$ is $\angle C = 72^{\circ}$, and the included angle for $\overline{FD}$ in $\triangle FDE$ is $\angle E=65^{\circ}$.
Since $72^{\circ}>65^{\circ}$ (i.e., $\angle C>\angle E$).

Step2: Compare the lengths of $\overline{AB}$ and $\overline{FD}$

By the Hinge Theorem, the side opposite the larger included angle is longer. So, $\overline{AB}$ (opposite $\angle C$) is longer than $\overline{FD}$ (opposite $\angle E$).

Answer:

$\overline{AB}$ is longer than $\overline{FD}$.