Question

consider the triangle. which shows the order of the angles from smallest to largest?
triangle with vertices a, b, c: side ac = 12, side cb = 16, side ab = 22
options:

• angle a, angle b, angle c
• angle b, angle a, angle c
• angle b, angle c, angle a
• angle c, angle a, angle b

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the side opposite an angle, the larger the angle. So we first identify the sides opposite each angle.

• Side opposite angle \( A \): \( BC = 16 \)
• Side opposite angle \( B \): \( AC = 12 \)
• Side opposite angle \( C \): \( AB = 22 \)

Step2: Order the sides from smallest to largest

Order of sides (from smallest to largest): \( AC = 12 \), \( BC = 16 \), \( AB = 22 \)

Step3: Order the angles based on opposite sides

Since the angle opposite the smaller side is smaller, the order of angles (from smallest to largest) is the angle opposite the smallest side, then the angle opposite the next - smallest side, then the angle opposite the largest side.

• Angle opposite \( AC \) (smallest side) is angle \( B \)
• Angle opposite \( BC \) (next - smallest side) is angle \( A \)
• Angle opposite \( AB \) (largest side) is angle \( C \)

So the order of angles from smallest to largest is angle \( B \), angle \( A \), angle \( C \)

Answer:

angle B, angle A, angle C