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: Sum of triangle angles

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 angles: \(60 + (3x - 2) + (2x + 7) = 180\).

Step2: Solve for x

Simplify the equation: \(60 + 3x - 2 + 2x + 7 = 180\) → \(5x + 65 = 180\) → \(5x = 115\) → \(x = 23\).

Step3: Find angle measures

Calculate \(m\angle B = 3(23) - 2 = 67^\circ\), \(m\angle C = 2(23) + 7 = 53^\circ\). So angles are \(60^\circ\), \(67^\circ\), \(53^\circ\). The largest angle is \(67^\circ\) (angle B), so the side opposite angle B (side AC) is the longest. Wait, no: in triangle ABC, side opposite angle A is BC, opposite angle B is AC, opposite angle C is AB. Wait, largest angle is angle B (\(67^\circ\)), so side opposite angle B is AC? Wait, no, wait: angle A is \(60\), angle B is \(67\), angle C is \(53\). So largest angle is B, so side opposite B is AC? Wait, no, triangle notation: angle A is at vertex A, so side opposite angle A is BC, side opposite angle B is AC, side opposite angle C is AB. Wait, but let's check again. Wait, angle B is \(67\), which is the largest. So side opposite angle B is AC? Wait, no, maybe I mixed up. Wait, in triangle ABC, vertices are A, B, C. So side opposite angle A (at A) is BC, side opposite angle B (at B) is AC, side opposite angle C (at C) is AB. So largest angle is B (\(67^\circ\)), so side opposite B is AC? Wait, no, wait, angle A: \(60\), angle B: \(67\), angle C: \(53\). So angle B is the largest. Therefore, the side opposite angle B is AC? Wait, no, maybe the problem is labeled as triangle ABC, so side opposite angle B is AC? Wait, no, maybe I made a mistake. Wait, let's re-express: in triangle ABC, angle at A: \(60\), angle at B: \(67\), angle at C: \(53\). So the largest angle is B, so the side opposite angle B is AC? Wait, no, side opposite angle B is AC? Wait, no, side opposite angle A is BC, side opposite angle B is AC, side opposite angle C is AB. So since angle B is the largest, side AC (opposite angle B) is the longest? Wait, but maybe the problem's triangle is labeled with sides: side a is BC (opposite angle A), side b is AC (opposite angle B), side c is AB (opposite angle C). So largest angle is B, so side AC (side b) is the longest? Wait, but maybe the answer is AC? Wait, no, wait, maybe I messed up the angle calculation. Wait, angle A: 60, angle B: 3x - 2, angle C: 2x + 7. Sum: 60 + 3x - 2 + 2x + 7 = 5x + 65 = 180 → 5x = 115 → x = 23. Then angle B: 323 - 2 = 69 - 2 = 67, angle C: 223 + 7 = 46 + 7 = 53. So angles: 60, 67, 53. So largest angle is B (67), so side opposite B is AC. Wait, but in triangle notation, side opposite angle B is AC? Wait, yes. So the longest side is AC? Wait, but the problem says "Side ____ is the longest side". Wait, maybe the triangle is labeled with sides: AB, BC, AC. Wait, maybe I made a mistake. Wait, no, let's recall: in a triangle, the longest side is opposite the largest angle. So largest angle is B (67°), so side opposite angle B is AC. So the longest side is AC? Wait, but maybe the problem's triangle is named ABC, so side opposite angle B is AC. So the answer is AC? Wait, but let's check again. Wait, angle A: 60, angle B: 67, angle C: 53. So angle B is the largest, so side opposite angle B is AC. So the longest side is AC.

Answer:

AC