Question

triangle inequalities
in the triangles, ( qr = de ) and ( sr = fe ).
which statement about the sides must be true?
( qs lt de )
( qs = df )
( df lt qs )
( 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, 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.
In \(\triangle QSR\) and \(\triangle DFE\), \(QR = DE\) (given), \(SR=FE\) (given), and \(\angle R = 62^{\circ}\), \(\angle E=50^{\circ}\), so \(\angle R>\angle E\).

Step2: Compare the third sides

By the Hinge Theorem, since \(\angle R>\angle E\) and \(QR = DE\), \(SR = FE\), then \(QS>DF\) (which is equivalent to \(DF < QS\)).

Answer:

\(DF < QS\)