Question

in the triangles, $overline{bc} cong overline{de}$ and $overline{ac} cong overline{fe}$. if $mangle c$ is greater than $mangle e$, then $overline{ab}$ is _____ $overline{df}$. options: longer than, congruent to, the same length as, shorter than

Explanation:

Step1: Identify Triangle Parts

We have two triangles, \( \triangle ABC \) and \( \triangle DFE \). Given \( BC \cong DE \) (one pair of congruent sides, marked with one red tick) and \( AC \cong FE \) (another pair of congruent sides, marked with two red ticks). The included angles are \( \angle C \) (in \( \triangle ABC \)) and \( \angle E \) (in \( \triangle DFE \)).

Step2: Apply 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.

Here, \( AC \cong FE \), \( BC \cong DE \), and \( m\angle C > m\angle E \). By the Hinge Theorem, the third side \( AB \) (opposite \( \angle C \)) must be longer than the third side \( DF \) (opposite \( \angle E \)).

Answer:

longer than