Question

in the triangles, qr = de and sr = fe. which statement about the sides must be true? qs = df df < qs qs < de sr = de

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, and 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. In \(\triangle QRS\) and \(\triangle DEF\), we have \(QR = DE\), \(SR=FE\), and \(\angle R = 62^{\circ}\), \(\angle E=50^{\circ}\), and \(\angle R>\angle E\).

Step2: Determine the relationship between the third - sides

By the Hinge - Theorem, the side opposite the larger angle in \(\triangle QRS\) (which is \(QS\)) is longer than the side opposite the smaller angle in \(\triangle DEF\) (which is \(DF\)). So \(QS>DF\), or equivalently \(DF < QS\).

Answer:

DF < QS