Question

1. match the correct answers.

ac ≅ df, bc ≅ ef, and ab > de
according to the converse of the hinge theorem:

a. be
b. ∠f
c. ad
d. ∠c

Explanation:

Step1: Recall Hinge Theorem Converse

The Converse of the Hinge Theorem (SSS Inequality 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.

In $\triangle ABC$ and $\triangle DEF$:

• $\overline{AC} \cong \overline{DF}$ (given)
• $\overline{BC} \cong \overline{EF}$ (given)
• $\overline{AB} > \overline{DE}$ (given)

So, the included angle for $\overline{AC}$ and $\overline{BC}$ in $\triangle ABC$ is $\angle C$, and the included angle for $\overline{DF}$ and $\overline{EF}$ in $\triangle DEF$ is $\angle F$. By the Converse of the Hinge Theorem, since $\overline{AB} > \overline{DE}$, we have $\angle C > \angle F$? Wait, no—wait, the sides: $\overline{AC} \cong \overline{DF}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AB} > \overline{DE}$. So the included angle of the longer third side ($\overline{AB}$) is $\angle C$, and the included angle of the shorter third side ($\overline{DE}$) is $\angle F$. So $\angle C > \angle F$? Wait, but the options: we need to match. Wait, maybe the question is to match the angle. Wait, the problem says "According to the Converse of the Hinge Theorem: [match]". Wait, maybe the left blank is the angle in $\triangle ABC$ and the right blank is the angle in $\triangle DEF$? Wait, the options are a. $\overline{BE}$, b. $\angle F$, c. $\overline{AD}$, d. $\angle C$. Wait, maybe the first blank is $\angle C$ and the second is $\angle F$? Wait, no—wait, the Converse of the Hinge Theorem: if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. So here, $\triangle ABC$ has sides $\overline{AC} \cong \overline{DF}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AB} > \overline{DE}$. So the included angle for $\overline{AC}$ and $\overline{BC}$ is $\angle C$, and for $\overline{DF}$ and $\overline{EF}$ is $\angle F$. So by Converse Hinge Theorem, $\angle C > \angle F$. So if we need to match, maybe the first blank is $\angle C$ (option d) and the second is $\angle F$ (option b). Wait, but the problem's blanks: there are two blanks? Wait, the image shows two blanks. So let's check the options:

a. $\overline{BE}$ (a segment)
b. $\angle F$ (an angle)
c. $\overline{AD}$ (a segment)
d. $\angle C$ (an angle)

So the Converse of the Hinge Theorem here would relate the angles. Since $\overline{AC} \cong \overline{DF}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AB} > \overline{DE}$, then the included angle of the longer side ($\overline{AB}$) is $\angle C$, and the included angle of the shorter side ($\overline{DE}$) is $\angle F$. So $\angle C > \angle F$. So if the first blank is for the larger angle and the second for the smaller, then first is $\angle C$ (d) and second is $\angle F$ (b).

Step2: Match the Angles

From the Converse of the Hinge Theorem, with $\overline{AC} \cong \overline{DF}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AB} > \overline{DE}$, the included angle $\angle C$ (in $\triangle ABC$) is greater than the included angle $\angle F$ (in $\triangle DEF$). So the first blank (left) should be $\angle C$ (option d) and the second blank (right) should be $\angle F$ (option b).

Answer:

First blank: d. $\angle C$, Second blank: b. $\angle F$