Question

applying the converse of the hinge theorem
given \\( \overline{ac} \cong \overline{fe} \\) and \\( \overline{cb} \cong \overline{ed} \\) which statement is correct?
options:
angle c is smaller than angle e.
angle c is congruent to angle e.
angle a is larger than angle b.
angle c is larger than angle e.

Explanation:

Step1: Recall Hinge Theorem Converse

The Converse of the Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

Step2: Identify Congruent Sides and Third Sides

We know \( \overline{AC} \cong \overline{FE} \) and \( \overline{CB} \cong \overline{ED} \). The third side of \( \triangle ACB \) is \( AB = 15 \) in, and the third side of \( \triangle FED \) is \( FD = 11 \) in. Since \( AB>FD \) (15 > 11), the included angle for \( \triangle ACB \) (which is \( \angle C \)) and the included angle for \( \triangle FED \) (which is \( \angle E \)): by the Converse of Hinge Theorem, \( \angle C>\angle E \).

Answer:

Angle C is larger than angle E.