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: Recall 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 \(QR = DE\) and \(SR=FE\), and \(\angle R = 62^{\circ}\), \(\angle E=50^{\circ}\), and \(\angle R>\angle E\).

Step2: Apply the Hinge - Theorem

By the Hinge - Theorem, the side opposite \(\angle R\) in \(\triangle QRS\) (which is \(QS\)) and the side opposite \(\angle E\) in \(\triangle DEF\) (which is \(DF\)) satisfy the relationship \(QS>DF\), or \(DF < QS\).

Answer:

DF < QS