Question

list the angles of the triangle in order from smallest to largest. note that the figure is not drawn to scale. choose the correct order of the angles from smallest to largest. a. ∠l, ∠m, ∠k b. ∠m, ∠k, ∠l c. ∠m, ∠l, ∠k d. ∠k, ∠l, ∠m e. ∠l, ∠k, ∠m f. ∠k, ∠m, ∠l

Explanation:

Step1: Recall the triangle side - angle relationship

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

Step2: Compare the side lengths

We have side lengths \(3x\), \(5x\). Since \(3x<5x\) (assuming \(x > 0\)). Let's consider the triangle \(\triangle LMK\). The side opposite \(\angle K\) is \(3x\), the side opposite \(\angle L\) is \(5x\), and \(\angle M=106^{\circ}\) (given).

Step3: Order the angles

Since the side opposite \(\angle K\) (\(3x\)) is the shortest, \(\angle K\) is the smallest. The side opposite \(\angle L\) (\(5x\)) is longer than the side opposite \(\angle K\) but shorter than the side opposite \(\angle M\) (implied by the angle - side relationship and the fact that \(\angle M = 106^{\circ}\) is already a large angle). So, \(\angle K<\angle L<\angle M\)

Answer:

D. $\angle K$, $\angle L$, $\angle M$