Question

list the angle measures of △abc in order from smallest to largest. assume that v is a positive number.
(image of triangle abc with side ab labeled 43v, angle at a labeled 93°, side ac labeled 130v)

Explanation:

Step1: Recall Triangle Side-Angle Relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. So we first compare the lengths of the sides: \(43v\), \(130v\), and the side opposite \(\angle A = 93^\circ\) (let's call its length \(BC\), but we can compare \(43v\) and \(130v\) directly since \(v>0\)). Since \(43v < 130v\), the angle opposite \(43v\) (which is \(\angle C\)) is smaller than the angle opposite \(130v\) (which is \(\angle B\)). Also, \(\angle A = 93^\circ\), we need to check the sum of angles in a triangle (\(180^\circ\)) to confirm, but first, let's analyze the sides.

Step2: Identify Angles Opposite Sides

• Side \(AB = 43v\) is opposite \(\angle C\).
• Side \(AC = 130v\) is opposite \(\angle B\).
• Side \(BC\) is opposite \(\angle A = 93^\circ\).

Since \(43v < 130v\), \(\angle C < \angle B\). Now, we know the sum of angles in a triangle is \(180^\circ\), so \(\angle A + \angle B + \angle C = 180^\circ\), so \(\angle B + \angle C = 180 - 93 = 87^\circ\). Since \(\angle B + \angle C = 87^\circ\), both \(\angle B\) and \(\angle C\) are less than \(93^\circ\), and since \(\angle C < \angle B\) (from side lengths), the order from smallest to largest is \(\angle C\), \(\angle B\), \(\angle A\).

Answer:

\(\angle C\), \(\angle B\), \(93^\circ\) (or \(\angle A\))