Question

1. match the correct answer.

ac ≅ de, bc ≅ ef, m∠e > m∠c
according to the hinge theorem:

a. m∠b
b. ab
c. df
d. m∠a

1. match the correct answer.

Explanation:

Step1: Recall Hinge Theorem

The Hinge Theorem (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, then the third side of the first triangle is longer than the third side of the second.

Step2: Identify Triangles and Sides/Angles

We have \(\triangle ABC\) and \(\triangle DEF\) (assuming \(D, E, F\) form the second triangle). Given \(\overline{AC} \cong \overline{DE}\), \(\overline{BC} \cong \overline{EF}\), and \(m\angle E > m\angle C\). The included angles are \(\angle C\) (in \(\triangle ABC\)) and \(\angle E\) (in \(\triangle DEF\)). The third sides are \(AB\) (in \(\triangle ABC\)) and \(DF\) (in \(\triangle DEF\))? Wait, no—wait, in \(\triangle ABC\), sides \(AC\) and \(BC\) with included angle \(\angle C\) have third side \(AB\). In \(\triangle DEF\), sides \(DE\) and \(EF\) with included angle \(\angle E\) have third side \(DF\). Wait, no, the Hinge Theorem says that if two sides are congruent (\(AC \cong DE\), \(BC \cong EF\)) and included angle \(\angle E > \angle C\), then the third side opposite the larger angle: the third side of the triangle with the larger included angle is longer. So, since \(m\angle E > m\angle C\), the third side of \(\triangle DEF\) (which is \(DF\)) should be longer than the third side of \(\triangle ABC\) (which is \(AB\))? Wait, no, wait: \(\triangle ABC\): sides \(AC\) and \(BC\), included angle \(\angle C\), third side \(AB\). \(\triangle DEF\): sides \(DE\) (congruent to \(AC\)) and \(EF\) (congruent to \(BC\)), included angle \(\angle E\) (larger than \(\angle C\)), third side \(DF\). So by Hinge Theorem, since \(\angle E > \angle C\), then \(DF > AB\)? Wait, no, the Hinge Theorem is: if in two triangles, two sides are congruent, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. So here, \(\triangle DEF\) has included angle \(\angle E\) (larger than \(\angle C\) in \(\triangle ABC\)), so third side \(DF\) (of \(\triangle DEF\)) is longer than third side \(AB\) (of \(\triangle ABC\)). So the first dropdown (left) should be \(DF\) (option c) and the second dropdown (right) should be \(AB\) (option b). Wait, but let's check the options again. The options are: a. \(m\angle B\), b. \(AB\), c. \(DF\), d. \(m\angle A\). So the inequality is \(DF > AB\) (since \(\angle E > \angle C\), so the third side of the triangle with larger included angle is longer). So the first blank (left of >) is \(DF\) (c) and the second blank (right of >) is \(AB\) (b). Wait, but let's confirm the Hinge Theorem again. The Hinge Theorem: If \( \overline{AC} \cong \overline{DE} \), \( \overline{BC} \cong \overline{EF} \), and \( m\angle C < m\angle E \), then \( AB < DF \) (or \( DF > AB \)). Yes, that's correct. So the correct match is \( DF > AB \), so left dropdown: c. \(DF\), right dropdown: b. \(AB\).

Answer:

Left dropdown: c. \(DF\)
Right dropdown: b. \(AB\)