Question

in the triangles, hg = mp and gk = pn. which statement about the sides and angles is true?
options:

• ( mangle g > mangle p )
• ( mangle p > mangle g )
• ( hk = mn )
• ( hg = pn )

(diagrams: triangle ( hgk ) with ( hk = 15 ) mm, ( gk ) with two congruence marks; triangle ( mpn ) with ( mn = 11 ) mm, ( pn ) with two congruence marks, ( mp ) with one congruence mark.)

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 third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

We know that \( HG = MP \) and \( GK = PN \). Now, let's look at the third sides: \( HK = 15 \) mm and \( MN = 11 \) mm. So, \( HK>MN \).

Step2: Apply the Hinge Theorem

The included angle for the sides \( HG \) and \( GK \) is \( \angle G \), and the included angle for the sides \( MP \) and \( PN \) is \( \angle P \). Since \( HK > MN \) (the third sides) and the other two sides are congruent (\( HG = MP \), \( GK = PN \)), by the Hinge Theorem, \( m\angle G>m\angle P \).

Let's check the other options:

• For \( m\angle P>m\angle G \): This is false because we just proved \( m\angle G > m\angle P \).
• For \( HK = MN \): \( HK = 15 \) mm and \( MN = 11 \) mm, so this is false.
• For \( HG = PN \): We know \( HG = MP \) and \( GK = PN \), there's no information to suggest \( HG = PN \), so this is false.

Answer:

\( m\angle G>m\angle P \) (the first option)