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: 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 congruent sides and included angles

We know that \(\overline{AC} \cong \overline{FE}\) and \(\overline{BC} \cong \overline{DE}\). The included angle for \(\triangle ABC\) at \(C\) is \(72^\circ\), and the included angle for \(\triangle FDE\) at \(E\) is \(65^\circ\).

Step3: Apply the Hinge Theorem

Since two sides of \(\triangle ABC\) (\(\overline{AC}\) and \(\overline{BC}\)) are congruent to two sides of \(\triangle FDE\) (\(\overline{FE}\) and \(\overline{DE}\)) respectively, and the included angle \( \angle C = 72^\circ\) is greater than the included angle \( \angle E = 65^\circ\), by the Hinge Theorem, the third side opposite the larger angle (which is \(\overline{AB}\) in \(\triangle ABC\) and \(\overline{FD}\) in \(\triangle FDE\)) will have \(\overline{AB}\) longer than \(\overline{FD}\)? Wait, no, wait. Wait, let's check the triangles again. Wait, in \(\triangle ABC\), the sides are \(AC\), \(BC\), and \(AB\). In \(\triangle FDE\), the sides are \(FE\), \(DE\), and \(FD\). Given \(AC \cong FE\), \(BC \cong DE\), and the included angles at \(C\) and \(E\) are \(72^\circ\) and \(65^\circ\) respectively. So the included angle for \(\triangle ABC\) is \( \angle C\) between \(AC\) and \(BC\), and for \(\triangle FDE\) is \( \angle E\) between \(FE\) and \(DE\). Since \( \angle C = 72^\circ> \angle E = 65^\circ\), then the side opposite? Wait, no, the Hinge Theorem says that if two sides are congruent, and the included angle is larger, then the third side is longer. So in \(\triangle ABC\), the third side is \(AB\), and in \(\triangle FDE\), the third side is \(FD\). Wait, but wait, is the included angle between the two congruent sides? Yes, \(AC\) and \(BC\) have included angle \(C\), \(FE\) and \(DE\) have included angle \(E\). So since \(AC \cong FE\), \(BC \cong DE\), and \( \angle C > \angle E\), then \(AB > FD\)? Wait, but let's check the labels again. Wait, the triangle on the left is \(A\), \(B\), \(C\) with \(AC\) horizontal, \(BC\) with a tick mark, \(AC\) with a tick mark? Wait, the diagram shows \(AC\) and \(FE\) with tick marks (so \(AC \cong FE\)), and \(BC\) and \(DE\) with tick marks (so \(BC \cong DE\)). So the two triangles have two sides congruent, and the included angles: \( \angle C = 72^\circ\), \( \angle E = 65^\circ\). So by Hinge Theorem, the side opposite? Wait, no, the Hinge Theorem is about the side opposite the included angle? Wait, no, the Hinge Theorem states that if in two triangles, 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. So in this case, first triangle is \(\triangle ABC\), second is \(\triangle FDE\). So \(AC \cong FE\), \(BC \cong DE\), included angle \( \angle C = 72^\circ\) (first triangle) and included angle \( \angle E = 65^\circ\) (second triangle). Since \(72^\circ>65^\circ\), then the third side of the first triangle (\(AB\)) is longer than the third side of the second triangle (\(FD\))? Wait, but that contradicts my initial thought. Wait, no, let's draw the triangles mentally. If two sides are equal, and the angle between them is larger, the third side is longer. So if you have two triangles with two sides equal, and one has a bigger angle between those two sides, the third side is longer. So in \(\triangle ABC\), angle at \(C\) is bigger, so \(AB\) is longer than \(FD\)? Wait, but the options include " \(\overline{AB}\) is longer than \(\overline{FD}\)". Wait, but let's check the angles again. Wait, maybe I mixed up the triangles. Wait, the triangle on the right is \(F\), \(D\), \(E\) with \(FE\) horizontal, \(DE\) with a tick mark, \(FE\) with a tick mark. So \(AC \cong FE\) (tick marks), \(BC \cong DE\) (tick marks). So the two triangles: \(\triangle ABC\) has sides \(AC\), \(BC\), \(AB\); \(\triangle FDE\) has sides \(FE\), \(DE\), \(FD\). So the included angles are at \(C\) (between \(AC\) and \(BC\)) and at \(E\) (between \(FE\) and \(DE\)). So \( \angle C = 72^\circ\), \( \angle E = 65^\circ\). Since \( \angle C > \angle E\), then by Hinge Theorem, \(AB > FD\)? Wait, but that would mean \(AB\) is longer than \(FD\), which is one of the options. Wait, but let's confirm the Hinge Theorem again. The Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. Yes, that's correct. So since \(AC \cong FE\), \(BC \cong DE\), and \( \angle C > \angle E\), then \(AB > FD\), so \(\overline{AB}\) is longer than \(\overline{FD}\). Wait, but wait, maybe I made a mistake in the triangle labels. Let's check the vertices. In \(\triangle ABC\), the vertices are \(A\), \(B\), \(C\). In \(\triangle FDE\), the vertices are \(F\), \(D\), \(E\). So the sides: \(AC\) is from \(A\) to \(C\), \(FE\) is from \(F\) to \(E\) (congruent). \(BC\) is from \(B\) to \(C\), \(DE\) is from \(D\) to \(E\) (congruent). So the included angles: at \(C\) (between \(A\) - \(C\) - \(B\)) and at \(E\) (between \(F\) - \(E\) - \(D\)). So yes, \( \angle C = 72^\circ\), \( \angle E = 65^\circ\). So since the included angle in \(\triangle ABC\) is larger, the third side \(AB\) is longer than the third side \(FD\) in \(\triangle FDE\). So the correct statement is " \(\overline{AB}\) is longer than \(\overline{FD}\)".

Answer:

\(\overline{AB}\) is longer than \(\overline{FD}\) (the option: \(\overline{AB}\) is longer than \(\overline{FD}\))