Question

in the triangles, $overline{ac}congoverline{fe}$ and $overline{bc}congoverline{de}$. which statement correctly compares $overline{ab}$ and $overline{fd}$? $overline{ab}$ and $overline{fd}$ are the same length. $overline{ab}$ is longer than $overline{fd}$. $overline{ab}$ is shorter than $overline{fd}$. $overline{ab}$ is longer than or the same length as $overline{fd}$

Explanation:

Step1: Recall the Hinge - Theorem

If two sides of one triangle are congruent to two sides of another triangle, and 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.

Step2: Identify congruent sides and included angles

In \(\triangle ABC\) and \(\triangle FDE\), we have \(\overline{AC}\cong\overline{FE}\) and \(\overline{BC}\cong\overline{DE}\). The included - angle for \(\overline{AC}\) and \(\overline{BC}\) in \(\triangle ABC\) is \(\angle C = 72^{\circ}\), and the included - angle for \(\overline{FE}\) and \(\overline{DE}\) in \(\triangle FDE\) is \(\angle E=65^{\circ}\). Since \(72^{\circ}>65^{\circ}\).

Step3: Apply the Hinge - Theorem

By the Hinge - Theorem, the side opposite the larger angle in \(\triangle ABC\) (which is \(\overline{AB}\)) is longer than the side opposite the smaller angle in \(\triangle FDE\) (which is \(\overline{FD}\)).

Answer:

\(\overline{AB}\) is longer than \(\overline{FD}\)