Question

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

Explanation:

Step1: Recall the Converse of the Hinge Theorem

The Converse of 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 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.

Step2: Identify the congruent sides and the third sides

We are given that \( GK = XZ \) and \( GH = XY \). So, two sides of \( \triangle GHK \) ( \( GH \) and \( GK \)) are congruent to two sides of \( \triangle XYZ \) ( \( XY \) and \( XZ \)). The third sides are \( HK = 21 \, \text{cm} \) and \( YZ = 26 \, \text{cm} \). Since \( 21 < 26 \) (i.e., \( HK < YZ \)).

Step3: Apply the Converse of the Hinge Theorem

By the Converse of the Hinge Theorem, since \( HK < YZ \) and \( GH = XY \), \( GK = XZ \), the included angle \( \angle G \) (between \( GH \) and \( GK \)) must be less than the included angle \( \angle X \) (between \( XY \) and \( XZ \)). So, \( m\angle G < m\angle X \).

Let's also check the other options:

• Option \( m\angle G>m\angle X \): This is false because we just proved \( m\angle G < m\angle X \).
• Option \( HG = XZ \): We know \( GK = XZ \) and \( GH = XY \), but there is no information to suggest \( HG = XZ \), so this is false.
• Option \( HK = YZ \): We know \( HK = 21 \, \text{cm} \) and \( YZ = 26 \, \text{cm} \), so \( HK

eq YZ \), this is false.

Answer:

\( m\angle G < m\angle X \) (the option with the statement \( m\angle G < m\angle X \))