Question

applying the hinge theorem
given ( ac = ln ) and ( ba = ml ), which statement must be true?
( bc>mn )
( ba = ln )
( bc < mn )
( bc = mn )

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 smaller than the included angle of the second triangle, then the third side of the first triangle is shorter than the third side of the second triangle.

Step2: Identify the sides and angles

We are given \(AC = LN\) and \(BA = ML\). The included angle in \(\triangle ABC\) is \(58^{\circ}\) (angle \(A\)) and the included angle in \(\triangle MLN\) is \(78^{\circ}\) (angle \(L\)). Since \(58^{\circ}<78^{\circ}\).

Step3: Apply the Hinge Theorem

By the Hinge Theorem, if \(AC = LN\), \(BA = ML\) and \(\angle A<\angle L\), then \(BC < MN\)

Answer:

\(BC < MN\)