Question

in the triangles, hg = mp and gk = pn. which statement about the sides and angles is true? o m∠g > m∠p o m∠p > m∠g o hk = mn o hg = pn

Explanation:

Step1: Apply 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 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.
We know that \(HG = MP\) and \(GK=PN\). Also, \(HK = 15\mathrm{mm}\) and \(MN = 11\mathrm{mm}\), so \(HK>MN\).

Step2: Analyze the angles

Since \(HK>MN\) (where \(HK\) and \(MN\) are the third sides of the two triangles \(\triangle HGK\) and \(\triangle MPN\) respectively) and \(HG = MP\), \(GK = PN\), by the Hinge Theorem, the angle opposite the longer side (\(HK\)) is larger. So \(m\angle G>m\angle P\).

Step3: Check other options

• For \(HK = MN\): \(HK = 15\mathrm{mm}\) and \(MN=11\mathrm{mm}\), so \(HK

eq MN\).

• For \(HG = PN\): Given \(HG = MP\) and \(GK = PN\), not \(HG = PN\).
• For \(m\angle P>m\angle G\): Contradicts the Hinge - Theorem result.

Answer:

\(m\angle G>m\angle P\)