Question

list the sides of each triangle in order from shortest to longest. δmkl with \angle m = 65^\circ, \angle k = 56^\circ, and \angle l = 59^\circ choose the correct order of the sides from shortest to longest.
o a. \overline{lk}, \overline{lm}, \overline{mk}
o b. \overline{lm}, \overline{mk}, \overline{lk}
o c. \overline{lk}, \overline{mk}, \overline{lm}
o d. \overline{lm}, \overline{lk}, \overline{mk}
o e. \overline{mk}, \overline{lk}, \overline{lm}
o f. \overline{mk}, \overline{lm}, \overline{lk}

Explanation:

Step1: Recall the triangle side - angle relationship

In a triangle, the larger the angle, the longer the side opposite it.

Step2: Compare the angles

We have \(\angle K = 56^{\circ}\), \(\angle L=59^{\circ}\), \(\angle M = 65^{\circ}\). So, \(\angle K<\angle L<\angle M\).

Step3: Determine the sides opposite the angles

The side opposite \(\angle K\) is \(\overline{LM}\), the side opposite \(\angle L\) is \(\overline{MK}\), and the side opposite \(\angle M\) is \(\overline{LK}\).
Since \(\angle K<\angle L<\angle M\), the sides (opposite to these angles) follow the order \(\overline{LK}<\overline{LM}<\overline{MK}\) (because of the side - angle relationship in a triangle: \(a = 2R\sin A\), \(b = 2R\sin B\), \(c = 2R\sin C\) where \(a,b,c\) are sides and \(A,B,C\) are angles opposite to them respectively and \(R\) is the circum - radius of the triangle. As the sine function is an increasing function for angles in \((0,\pi)\)).

Answer:

A. \(\overline{LK}\), \(\overline{LM}\), \(\overline{MK}\)