Question

select the correct answer.
a building has a triangular rooftop terrace which is modeled by triangle abc. in triangle abc, the measure of ∠a is 55°, the measure of ∠b is 65°, and the measure of ∠c is 60°. which side of the terrace has the greatest length?
a. cannot be determined
b. (overline{bc})
c. (overline{ab})
d. (overline{ac})

Explanation:

Step1: Find the third angle

In a triangle, the sum of interior angles is \(180^\circ\). Given \(\angle A = 55^\circ\), \(\angle B = 65^\circ\), so \(\angle C=180^\circ - 55^\circ - 65^\circ = 60^\circ\).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the side opposite to it. Compare the angles: \(\angle B = 65^\circ\), \(\angle C = 60^\circ\), \(\angle A = 55^\circ\). So \(\angle B\) is the largest angle. The side opposite \(\angle B\) is \(\overline{AC}\)? Wait, no: opposite \(\angle A\) is \(\overline{BC}\), opposite \(\angle B\) is \(\overline{AC}\), opposite \(\angle C\) is \(\overline{AB}\). Wait, no, let's correct: In \(\triangle ABC\), side opposite \(\angle A\) is \(BC\), opposite \(\angle B\) is \(AC\), opposite \(\angle C\) is \(AB\). Wait, no, angle \(A\) is at vertex \(A\), so side opposite is \(BC\) (connecting \(B\) and \(C\)). Angle \(B\) is at vertex \(B\), side opposite is \(AC\) (connecting \(A\) and \(C\)). Angle \(C\) is at vertex \(C\), side opposite is \(AB\) (connecting \(A\) and \(B\)). Now, angles: \(\angle A = 55^\circ\), \(\angle B = 65^\circ\), \(\angle C = 60^\circ\). So the largest angle is \(\angle B = 65^\circ\), so the side opposite \(\angle B\) (which is \(AC\))? Wait, no, wait the options: Option C is \(\overline{AB}\), Option B is \(\overline{BC}\), Option D is \(\overline{AC}\). Wait, let's re - check: \(\angle A = 55^\circ\), opposite side \(BC\); \(\angle B = 65^\circ\), opposite side \(AC\); \(\angle C = 60^\circ\), opposite side \(AB\). So the largest angle is \(\angle B = 65^\circ\), so the longest side is opposite \(\angle B\), which is \(AC\)? Wait, no, the options: Wait the question is which side has the greatest length. Wait, maybe I mixed up. Wait, \(\angle B = 65^\circ\) is the largest angle, so the side opposite to \(\angle B\) is \(AC\) (since in \(\triangle ABC\), side opposite \(\angle B\) is \(AC\)). Wait, but let's check the angle measures again: \(\angle A = 55\), \(\angle B = 65\), \(\angle C = 60\). So order of angles: \(\angle B>\angle C>\angle A\). So the sides opposite: side opposite \(\angle B\) is \(AC\), side opposite \(\angle C\) is \(AB\), side opposite \(\angle A\) is \(BC\). So the longest side is \(AC\) (opposite the largest angle \(\angle B\)). Wait, but let's check the options: Option D is \(\overline{AC}\). Wait, but maybe I made a mistake. Wait, let's list the sides and their opposite angles:

• Side \(BC\): opposite \(\angle A\) (\(55^\circ\))
• Side \(AC\): opposite \(\angle B\) (\(65^\circ\))
• Side \(AB\): opposite \(\angle C\) (\(60^\circ\))

Since \(65^\circ\) (angle \(B\)) is the largest angle, the side opposite to it (\(AC\)) is the longest side. Wait, but let's check the options again. The options are:

A. Cannot be determined

B. \(\overline{BC}\)

C. \(\overline{AB}\)

D. \(\overline{AC}\)

So according to the angle - side relationship (larger angle opposite longer side), since \(\angle B = 65^\circ\) is the largest angle, the side opposite \(\angle B\) is \(\overline{AC}\), so \(\overline{AC}\) is the longest side.

Answer:

D. \(\overline{AC}\)