Question

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

Explanation:

Step1: Recall the Hinge Theorem

The Hinge Theorem (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.

Step2: Identify Congruent Sides and Angles

• Given \( AC = LN \) and \( BA = ML \).
• The included angle for \( \triangle ABC \) at \( A \) is \( \angle A = 58^\circ \), and the included angle for \( \triangle LMN \) at \( L \) is \( \angle L = 78^\circ \).
• Since \( 58^\circ< 78^\circ \), by the Hinge Theorem, the third side opposite the smaller angle ( \( BC \) in \( \triangle ABC \)) is shorter than the third side opposite the larger angle ( \( MN \) in \( \triangle LMN \)). So \( BC < MN \).

Answer:

BC < MN (the option with "BC < MN")