Question

in the triangles, \\( \overline { b c } \cong \overline { d e } \\) and \\( \overline { a c } \cong \overline { f e } \\). if \\( m \angle c \\) is greater than \\( m \angle e \\), then \\( \overline { a b } \\) is \\( \overline { d f } \\). congruent to longer than shorter than the same length as

Explanation:

Step1: Recall the Hinge Theorem

The Hinge Theorem (also known as the SAS Inequality 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.

Step2: Identify the triangles and sides/angles

We have two triangles: $\triangle ABC$ and $\triangle DFE$. We know that $\overline{BC} \cong \overline{DE}$ and $\overline{AC} \cong \overline{FE}$. The included angles are $\angle C$ (in $\triangle ABC$) and $\angle E$ (in $\triangle DFE$). We are given that $m\angle C > m\angle E$.

Step3: Apply the Hinge Theorem

By the Hinge Theorem, since two sides of $\triangle ABC$ ($\overline{AC}$ and $\overline{BC}$) are congruent to two sides of $\triangle DFE$ ($\overline{FE}$ and $\overline{DE}$) respectively, and the included angle $\angle C$ is larger than the included angle $\angle E$, the third side of $\triangle ABC$ (which is $\overline{AB}$) must be longer than the third side of $\triangle DFE$ (which is $\overline{DF}$).

Answer:

longer than