Question

in the triangles, \\(\overline{gk} \cong \overline{pn}\\) and \\(\overline{hg} \cong \overline{mp}\\). which statement correctly compares the angles?
(images of two triangles: triangle ghk with gh marked congruent to gk? wait, no, the first triangle has vertices g, h, k: gh has one tick, gk has two ticks, hk is 32 cm. second triangle has vertices p, m, n: pm has one tick, pn has two ticks, mn is 40 cm. then four options:

1. angle g is larger than angle p.
2. angle g is congruent to angle n.
3. angle g is smaller than angle p.
4. angle g is congruent to angle p.)

Explanation:

Step1: Recall the Hinge Theorem

The Hinge 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 smaller than the included angle of the second triangle, then the third side of the first triangle is shorter than the third side of the second triangle. Conversely, if the third side of one triangle is shorter than the third side of another triangle (with two congruent sides), the included angle of the first triangle is smaller than the included angle of the second triangle.

Step2: Identify Congruent Sides and Third Sides

We know \( \overline{GK} \cong \overline{PN} \) and \( \overline{HG} \cong \overline{MP} \). The third sides are \( \overline{HK} = 32 \, \text{cm} \) and \( \overline{MN} = 40 \, \text{cm} \). Since \( 32 < 40 \), the included angle for the triangle with the shorter third side (triangle \( GHK \)) will be smaller. The included angle for \( \triangle GHK \) is \( \angle G \), and for \( \triangle PMN \) is \( \angle P \). So \( \angle G < \angle P \), meaning \( \angle G \) is smaller than \( \angle P \).

Answer:

Angle G is smaller than angle P.