Question

1. choose the correct answer.

$overline{gi} \cong overline{jl}$, $overline{gh} \cong overline{jk}$, and $hi > kl$

according to the converse of the hinge theorem:

a. $m\angle j$
b. $m\angle g$
c. $m\angle i$
d. $m\angle l

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 sides are not congruent, then the larger third side is opposite the larger included angle.

Step2: Identify Congruent Sides and Included Angles

We have \( \overline{GI} \cong \overline{JL} \), \( \overline{GH} \cong \overline{JK} \), and \( HI > KL \). The included angles for the congruent sides are \( \angle G \) (in \( \triangle GHI \)) and \( \angle J \) (in \( \triangle JKL \)). Since \( HI > KL \), the angle opposite \( HI \) (which is \( \angle G \)) should be larger than the angle opposite \( KL \) (which is \( \angle J \))? Wait, no—wait, let's correct. Wait, the two sides: \( GI \cong JL \), \( GH \cong JK \), so the included angles are \( \angle G \) (between \( GH \) and \( GI \)) and \( \angle J \) (between \( JK \) and \( JL \)). The third sides are \( HI \) and \( KL \). Since \( HI > KL \), by Converse Hinge Theorem, the angle opposite \( HI \) (which is \( \angle G \)) is larger than the angle opposite \( KL \) (which is \( \angle J \))? Wait, no, wait: the included angle is between the two congruent sides. Wait, \( GH \cong JK \), \( GI \cong JL \), so the included angles are \( \angle G \) (for \( \triangle GHI \)) and \( \angle J \) (for \( \triangle JKL \)). The third sides are \( HI \) and \( KL \). Since \( HI > KL \), then \( m\angle G > m\angle J \)? Wait, no, wait the Converse: if two sides congruent, third side longer implies included angle larger. Wait, let's rephrase: In \( \triangle GHI \) and \( \triangle JKL \), \( GH = JK \), \( GI = JL \), \( HI > KL \). So the included angle for \( \triangle GHI \) is \( \angle G \), for \( \triangle JKL \) is \( \angle J \). Then by Converse Hinge Theorem, \( m\angle G > m\angle J \). Wait, but the options are about which angle is larger? Wait, the question is probably asking which angle is larger, and the options are \( m\angle J \), \( m\angle G \), \( m\angle I \), \( m\angle L \). Wait, maybe I misread. Wait, the problem says "According to the Converse of the Hinge Theorem: [first blank] > [second blank]". Wait, the first blank options are a - d, second blank too? Wait, the original problem has two dropdowns, but the options are a: \( m\angle J \), b: \( m\angle G \), c: \( m\angle I \), d: \( m\angle L \). Wait, let's re-express.

Wait, the two triangles: \( \triangle GHI \) and \( \triangle JKL \). Sides: \( GH \cong JK \), \( GI \cong JL \), \( HI > KL \). The included angles are \( \angle G \) (between \( GH \) and \( GI \)) and \( \angle J \) (between \( JK \) and \( JL \)). By Converse Hinge Theorem, since \( HI > KL \), the included angle opposite the longer side (HI) is larger. So \( m\angle G > m\angle J \). So the first angle (larger) is \( m\angle G \) (option b) and the second is \( m\angle J \) (option a). Wait, but the problem's dropdowns: first dropdown options are a - d, second too. So the correct comparison is \( m\angle G > m\angle J \), so first blank is \( m\angle G \) (option b) and second is \( m\angle J \) (option a). But let's confirm.

Wait, Hinge Theorem Converse: If two sides of one triangle are congruent to two sides of another triangle, and the third sides are unequal, then the larger third side is opposite the larger included angle. So in \( \triangle GHI \) and \( \triangle JKL \):

- \( GH \cong JK \) (side 1)
- \( GI \cong JL \) (side 2)
- \( HI > KL \) (side 3)

Included angles: \( \angle G \) (between side1 and side2 in \( \triangle GHI \)) and \( \angle J \) (between side1 and side2 in \( \triangle JKL \)).

Since \( HI > KL \), the angle opposite \( HI \) (which is \( \angle G \)) is larger than the angle opposite \( KL \) (which is \( \angle J \)). So \( m\angle G > m\angle J \).

So the first dropdown (larger angle) is \( m\angle G \) (option b) and the second (smaller angle) is \( m\angle J \) (option a).

Answer:

First blank: b. \( m\angle G \), Second blank: a. \( m\angle J \)