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. (overline{ac})
b. (overline{bc})
c. (overline{ab})
d. cannot be determined

Question

Accepted Answer

# Explanation:
## Step1: Recall triangle angle - side relation
In a triangle, the larger the angle, the longer the side opposite to it. The formula for the sum of interior angles of a triangle is $ \angle A+\angle B+\angle C = 180^{\circ}$, but here we can directly use the angle - side opposite relationship: side $a$ is opposite $ \angle A$, side $b$ is opposite $ \angle B$, side $c$ is opposite $ \angle C$, and if $ \angle X>\angle Y$, then the side opposite $ \angle X$ is longer than the side opposite $ \angle Y$.

## Step2: Identify angles and their opposite sides
- For $ \angle A = 55^{\circ}$, the side opposite to it is $BC$ (since in $ \triangle ABC$, side opposite $ \angle A$ is $BC$).
- For $ \angle B=65^{\circ}$, the side opposite to it is $AC$ (side opposite $ \angle B$ is $AC$).
- For $ \angle C = 60^{\circ}$, the side opposite to it is $AB$ (side opposite $ \angle C$ is $AB$).

## Step3: Compare the angles
Now, compare the measures of the angles: $ \angle B=65^{\circ}$, $ \angle C = 60^{\circ}$, $ \angle A=55^{\circ}$. So, $ \angle B>\angle C>\angle A$.

## Step4: Determine the longest side
Since $ \angle B$ is the largest angle among $ \angle A$, $ \angle B$, and $ \angle C$, the side opposite to $ \angle B$ (which is $AC$) is not the longest? Wait, no, wait: Wait, let's re - check. Wait, $ \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 $AC$? No, wait, in triangle $ABC$:
- Vertex $A$, opposite side $BC$
- Vertex $B$, opposite side $AC$
- Vertex $C$, opposite side $AB$

Wait, no, I made a mistake. Let's correct: In $ \triangle ABC$, the side opposite $ \angle A$ is $BC$ (because $ \angle A$ is at vertex $A$, and the side opposite is $BC$), side opposite $ \angle B$ is $AC$ (at vertex $B$, opposite side is $AC$), side opposite $ \angle C$ is $AB$ (at vertex $C$, opposite side is $AB$).

Now, $ \angle B = 65^{\circ}$, $ \angle C=60^{\circ}$, $ \angle A = 55^{\circ}$. So $ \angle B$ is the largest angle. So the side opposite $ \angle B$ (which is $AC$)? Wait, no, wait, no: Wait, if $ \angle B$ is at vertex $B$, then the side opposite is $AC$. Wait, but let's check the angles again. Wait, $ \angle B = 65^{\circ}$, $ \angle C = 60^{\circ}$, $ \angle A=55^{\circ}$. So the largest angle is $ \angle B$ with measure $65^{\circ}$. So the side opposite $ \angle B$ is $AC$? No, wait, no, I think I mixed up. Wait, let's take an example: In triangle $ABC$, angle at $A$: $ \angle A$, side $BC$ is opposite. Angle at $B$: $ \angle B$, side $AC$ is opposite. Angle at $C$: $ \angle C$, side $AB$ is opposite.

So, $ \angle A = 55^{\circ}$, opposite side $BC$; $ \angle B=65^{\circ}$, opposite side $AC$; $ \angle C = 60^{\circ}$, opposite side $AB$.

Now, since $ \angle B=65^{\circ}$ is the largest angle, the side opposite to it ($AC$)? Wait, no, wait, $ \angle B$ is 65, $ \angle C$ is 60, $ \angle A$ is 55. So $ \angle B$ is the largest. So the side opposite $ \angle B$ is $AC$? Wait, no, that can't be. Wait, no, I think I made a mistake in the opposite sides. Let's label the triangle properly:

Let's denote:

- $ \angle A$ is at vertex $A$, between sides $AB$ and $AC$, so the side opposite $ \angle A$ is $BC$.

- $ \angle B$ is at vertex $B$, between sides $AB$ and $BC$, so the side opposite $ \angle B$ is $AC$.

- $ \angle C$ is at vertex $C$, between sides $AC$ and $BC$, so the side opposite $ \angle C$ is $AB$.

Now, comparing the angles: $ \angle B = 65^{\circ}$, $ \angle C=60^{\circ}$, $ \angle A = 55^{\circ}$. So $ \angle B$ is the largest angle. Therefore, the side opposite $ \angle B$ (which is $AC$)? Wait, no, wait, $ \angle B$ is 65, so the side opposite $ \angle B$ is $AC$. But wait, let's check the lengths. Wait, no, if $ \angle B$ is the largest angle, then the side opposite $ \angle B$ ( $AC$) should be the longest? But wait, let's re - calculate the angles. Wait, the sum of angles: $55 + 65+60=180$, which is correct.

Wait, but the options are $AC$, $BC$, $AB$, or cannot be determined. Wait, maybe I made a mistake in the opposite sides. Let's re - define:

In triangle $ABC$:

- Side $AB$: connects $A$ and $B$, opposite $ \angle C$

- Side $BC$: connects $B$ and $C$, opposite $ \angle A$

- Side $AC$: connects $A$ and $C$, opposite $ \angle B$

So, angle opposite $AB$ is $ \angle C$ (60°), angle opposite $BC$ is $ \angle A$ (55°), angle opposite $AC$ is $ \angle B$ (65°).

Since $ \angle B = 65^{\circ}$ is the largest angle, the side opposite $ \angle B$ (which is $AC$)? No, wait, $ \angle B$ is 65, so the side opposite $ \angle B$ is $AC$. But wait, the options:

Option A: $AC$

Option B: $BC$

Option C: $AB$

Option D: Cannot be determined.

Wait, but $ \angle B$ is the largest angle, so the side opposite $ \angle B$ ( $AC$) is the longest? Wait, no, wait, $ \angle B = 65^{\circ}$, so the side opposite $ \angle B$ is $AC$. So $AC$ is opposite $ \angle B$ (65°), $AB$ is opposite $ \angle C$ (60°), $BC$ is opposite $ \angle A$ (55°). So since $65^{\circ}>60^{\circ}>55^{\circ}$, the side opposite the largest angle ( $AC$) is the longest. So the longest side is $AC$, which is option A.

Wait, but let's check again. Let's list the angles and their opposite sides:

- $ \angle A = 55^{\circ}$, opposite side $BC$

- $ \angle B=65^{\circ}$, opposite side $AC$

- $ \angle C = 60^{\circ}$, opposite side $AB$

Since $ \angle B$ is the largest angle, the side opposite to it ( $AC$) is the longest side.

# Answer:
A. $\overline{AC}$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall triangle angle - side relation

Step2: Identify angles and their opposite sides

Step3: Compare the angles

Step4: Determine the longest side

Answer: