Question

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

Explanation:

Step1: Recall the Hinge Theorem

The Hinge Theorem (also known as the SAS Inequality 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.

In $\triangle ABC$ and $\triangle MLN$:

• We know that $AC = LN$ (given, marked with two ticks) and $BA = ML$ (given, marked with one tick).
• The included angle for $\triangle ABC$ at $A$ is $\angle A = 58^\circ$, and the included angle for $\triangle MLN$ at $L$ is $\angle L = 78^\circ$.

Step2: Apply the Hinge Theorem

Since $58^\circ<78^\circ$ (i.e., $\angle A < \angle L$), and two sides of $\triangle ABC$ ($AC$ and $BA$) are congruent to two sides of $\triangle MLN$ ($LN$ and $ML$) respectively, by the Hinge Theorem, the third side opposite the smaller included angle (which is $BC$ in $\triangle ABC$) must be shorter than the third side opposite the larger included angle (which is $MN$ in $\triangle MLN$). So, $BC < MN$.

Let's also check the other options:

• Option "BC > MN": This would be true if the included angle of the first triangle was larger, but here $\angle A < \angle L$, so this is false.
• Option "BC = MN": This would be true only if the included angles were equal (by SAS congruence), but the angles are not equal, so this is false.
• Option "BA = LN": We know $BA = ML$ and $AC = LN$, but there's no information to suggest $BA = LN$, so this is false.

Answer:

BC < MN (the first option: BC < MN)