Question

1. match the correct answers.

$overline{gi} cong overline{jl}, overline{gh} cong overline{jk},$ and $\angle g > \angle j$
according to the hinge theorem:

a. $hi$
b. $\angle l$
c. $\angle h$
d. $kl

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, 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.

Step2: Identify Corresponding Parts

We have \(\overline{GI} \cong \overline{JL}\) and \(\overline{GH} \cong \overline{JK}\). The included angles are \(\angle G\) (in \(\triangle GHI\)) and \(\angle J\) (in \(\triangle JKL\)). Given \(\angle G > \angle J\), by the Hinge Theorem, the third side of \(\triangle GHI\) (which is \(\overline{HI}\)) should be longer than the third side of \(\triangle JKL\) (which is \(\overline{KL}\)). So we need to match \(HI\) and \(KL\) with the inequality. Also, for the angles, if we consider the converse or other parts, but here the main sides from the Hinge Theorem application are \(HI\) and \(KL\) with \(HI > KL\). Wait, but the options: let's check the sides. The first blank (left) and second (right). So according to Hinge Theorem, since \(\angle G > \angle J\), \(HI > KL\). So the left blank should be \(HI\) (option a) and the right blank should be \(KL\) (option d)? Wait, no, maybe the other way. Wait, the triangles: \(\triangle GHI\) and \(\triangle JKL\). Sides \(GI = JL\), \(GH = JK\), included angle \(\angle G > \angle J\), so \(HI > KL\). So the first line (left) is \(HI\) (a) and the second (right) is \(KL\) (d)? Wait, but let's confirm the Hinge Theorem. Hinge Theorem: If \(AB = DE\), \(AC = DF\), and \(\angle A > \angle D\), then \(BC > EF\). So in our case, \(GI = JL\) (AB=DE), \(GH = JK\) (AC=DF), \(\angle G > \angle J\) (\(\angle A > \angle D\)), so \(HI > KL\) (BC > EF). So the first blank (left) is \(HI\) (a) and the second (right) is \(KL\) (d). Wait, but the options: a. \(HI\), d. \(KL\). So the matching is \(HI > KL\), so left: \(HI\) (a), right: \(KL\) (d).

Answer:

Left: a. \(HI\), Right: d. \(KL\)