Question

multiple choice
question 10
consider the diagram below. the image is not necessarily drawn to scale.
what is the order of the angle measures from least to greatest?
geometric figure
(diagram of triangle abc with side ab = 14, bc = 12, ac = 15)

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the length of a side, the larger the measure of the angle opposite that side. This is based on the fact that in a triangle, if one side is longer than another side, the angle opposite the longer side is larger.

Step2: Identify the sides and their opposite angles

• Side \( BC = 14 \), the angle opposite to \( BC \) is \( \angle A \).
• Side \( AC = 15 \), the angle opposite to \( AC \) is \( \angle B \).
• Side \( AB = 12 \), the angle opposite to \( AB \) is \( \angle C \).

Step3: Order the sides by length

First, we order the lengths of the sides from least to greatest: \( AB = 12\), \( BC = 14\), \( AC = 15 \).

Step4: Order the angles based on the side - angle relationship

Since the angle opposite a shorter side is smaller, the angles opposite these sides will follow the same order. The angle opposite \( AB \) (length 12) is \( \angle C \), the angle opposite \( BC \) (length 14) is \( \angle A \), and the angle opposite \( AC \) (length 15) is \( \angle B \). So the order of the angles from least to greatest is \( \angle C\), \( \angle A\), \( \angle B \).

Answer:

The order of the angle measures from least to greatest is \( \angle C<\angle A<\angle B \) (where \( \angle C \) is opposite side \( AB = 12 \), \( \angle A \) is opposite side \( BC = 14 \), and \( \angle B \) is opposite side \( AC = 15 \))