Question

applying the converse of the hinge theorem
given gk = xz and gh = xy, which statement must be true?
hg = xz
hk = yz
m∠g < m∠x
m∠g > m∠x

Explanation:

Step1: Recall the Hinge Theorem (SAS Inequality Theorem) and its Converse

The Hinge Theorem (also known as the 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 triangle, then the third side of the first triangle is longer than the third side of the second triangle. 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 side of the first triangle is shorter than the third side of the second triangle, then the included angle of the first triangle is smaller than the included angle of the second triangle.

In this problem, we have two triangles: \(\triangle GHK\) and \(\triangle XYZ\). We are given that \(GK = XZ\) (one pair of congruent sides) and \(GH = XY\) (another pair of congruent sides). We need to compare the included angles \(\angle G\) (in \(\triangle GHK\)) and \(\angle X\) (in \(\triangle XYZ\)) based on the lengths of the third sides \(HK\) and \(YZ\).

Step2: Analyze the lengths of the third sides

From the diagram, we can see that \(HK = 21\space\text{cm}\) and \(YZ = 26\space\text{cm}\). So, \(HK < YZ\) (since \(21 < 26\)).

Step3: Apply the Converse of the Hinge Theorem

Since \(GK = XZ\), \(GH = XY\) (two pairs of congruent sides), and \(HK < YZ\) (the third side of \(\triangle GHK\) is shorter than the third side of \(\triangle XYZ\)), by the converse of the Hinge Theorem, the included angle opposite the shorter third side (\(\angle G\) in \(\triangle GHK\)) must be smaller than the included angle opposite the longer third side (\(\angle X\) in \(\triangle XYZ\)). Therefore, \(m\angle G < m\angle X\).

Now let's check the other options:

• Option "HG = XZ": We are given \(GH = XY\), not \(HG = XZ\), so this is false.
• Option "HK = YZ": We saw that \(HK = 21\space\text{cm}\) and \(YZ = 26\space\text{cm}\), so \(HK

eq YZ\), this is false.

• Option "m\angle G > m\angle X": This is the opposite of what we concluded from the converse of the Hinge Theorem, so this is false.

Answer:

The statement that must be true is \(m\angle G < m\angle X\) (the option with \(m\angle G < m\angle X\)).