Question

type the correct answer in the box.
in triangle abc, which side is the longest if these are the measures of the angles?
m∠a = 60°, m∠b = (3x - 2)°, m∠c = (2x + 7)°
side is the longest side.

Explanation:

Step1: Use the angle - sum property of a triangle

The sum of angles in a triangle is \(180^{\circ}\). So, \(m\angle A+m\angle B + m\angle C=180^{\circ}\).
Substitute the given angle measures: \(60+(3x - 2)+(2x + 7)=180\).
Simplify the left - hand side: \(60+3x-2 + 2x+7=180\), which becomes \(5x+65 = 180\).
Subtract 65 from both sides: \(5x=180 - 65\), so \(5x=115\).
Divide both sides by 5: \(x=\frac{115}{5}=23\).

Step2: Find the measures of \(\angle B\) and \(\angle C\)

For \(\angle B\): Substitute \(x = 23\) into \(m\angle B=(3x - 2)^{\circ}\). Then \(m\angle B=(3\times23-2)^{\circ}=(69 - 2)^{\circ}=67^{\circ}\).
For \(\angle C\): Substitute \(x = 23\) into \(m\angle C=(2x + 7)^{\circ}\). Then \(m\angle C=(2\times23+7)^{\circ}=(46 + 7)^{\circ}=53^{\circ}\).

Step3: Use the side - angle relationship

In a triangle, the larger angle has the longer side opposite to it.
Since \(m\angle B = 67^{\circ}\), \(m\angle A=60^{\circ}\), and \(m\angle C = 53^{\circ}\), and \(67^{\circ}>60^{\circ}>53^{\circ}\).
The side opposite \(\angle B\) is \(AC\).

Answer:

\(AC\)