Question

1. suppose gf < yf in the figure. write an inequality that relates m∠fhy and m∠fhg.

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

In triangles \( \triangle FHY \) and \( \triangle FHG \) (assuming the figure has \( FH \) as a common side and \( HY = HG \) (marked with \( x \))), we have \( HY = HG \), \( FH = FH \) (common side), and \( GF < YF \).

Step2: Apply the Hinge Theorem

Since \( HY = HG \), \( FH = FH \), and \( GF < YF \), by the converse of the Hinge Theorem, the included angle opposite the shorter side will be smaller. The included angle for \( GF \) is \( m\angle FHG \) and the included angle for \( YF \) is \( m\angle FHY \). So, \( m\angle FHG < m\angle FHY \) or \( m\angle FHY > m\angle FHG \).

Answer:

\( m\angle FHY > m\angle FHG \) (or \( m\angle FHG < m\angle FHY \))