Question

consider the triangle.
image of triangle with vertices a (top), c (bottom left), b (bottom right); ac = 12, cb = 8, ab = 15
the measures of the angles of the triangle are 32°, 53°, 95°. based on the side lengths, what are the measures of each angle?
○ ( mangle a = 95^circ ), ( mangle b = 53^circ ), ( mangle c = 32^circ )
○ ( mangle a = 32^circ ), ( mangle b = 53^circ ), ( mangle c = 95^circ )
○ ( mangle a = 43^circ ), ( mangle b = 32^circ ), ( mangle c = 95^circ )
○ ( mangle a = 53^circ ), ( mangle b = 95^circ ), ( mangle c = 32^circ )

Explanation:

Step1: Recall the triangle side - angle relationship

In a triangle, the larger the side length, the larger the angle opposite to it. So we first identify the lengths of the sides: \(AC = 12\), \(BC = 8\), \(AB=15\). So the order of side lengths from longest to shortest is \(AB(15)>AC(12)>BC(8)\).

Step2: Determine the order of angles

The angle opposite to \(AB\) is \(\angle C\), the angle opposite to \(AC\) is \(\angle B\), and the angle opposite to \(BC\) is \(\angle A\). So the order of angles from largest to smallest should be \(\angle C>\angle B>\angle A\) (since the largest side is opposite the largest angle and vice - versa).

The given angles are \(32^{\circ}\), \(53^{\circ}\), \(95^{\circ}\). Arranging them from largest to smallest: \(95^{\circ}>53^{\circ}>32^{\circ}\). So \(\angle C = 95^{\circ}\), \(\angle B=53^{\circ}\), \(\angle A = 32^{\circ}\) does not match. Wait, no: Wait, the side \(AB = 15\) is opposite \(\angle C\), side \(AC = 12\) is opposite \(\angle B\), side \(BC = 8\) is opposite \(\angle A\). So the largest side \(AB = 15\) is opposite \(\angle C\), so \(\angle C\) should be the largest angle (\(95^{\circ}\)). The side \(AC = 12\) is opposite \(\angle B\), so \(\angle B\) is the middle - sized angle (\(53^{\circ}\)). The side \(BC = 8\) is opposite \(\angle A\), so \(\angle A\) is the smallest angle (\(32^{\circ}\))? No, wait, let's re - check. Wait, in triangle \(ABC\), side \(BC\) is between \(A\) and \(C\)? No, the vertices are \(A\), \(B\), \(C\). So side \(AB\) is between \(A\) and \(B\), side \(BC\) is between \(B\) and \(C\), side \(AC\) is between \(A\) and \(C\). So the angle at \(A\) (\(\angle A\)) is between sides \(AB\) and \(AC\), the angle at \(B\) (\(\angle B\)) is between sides \(AB\) and \(BC\), the angle at \(C\) (\(\angle C\)) is between sides \(AC\) and \(BC\).

So the length of side opposite \(\angle A\) is \(BC = 8\), opposite \(\angle B\) is \(AC = 12\), opposite \(\angle C\) is \(AB = 15\). So since \(AB>AC>BC\), then \(\angle C>\angle B>\angle A\). The angles are \(95^{\circ}\), \(53^{\circ}\), \(32^{\circ}\). So \(\angle C = 95^{\circ}\), \(\angle B = 53^{\circ}\), \(\angle A=32^{\circ}\) is not an option. Wait, the options:

Option 1: \(m\angle A = 95^{\circ}\), \(m\angle B = 53^{\circ}\), \(m\angle C=32^{\circ}\)

Option 2: \(m\angle A = 32^{\circ}\), \(m\angle B = 53^{\circ}\), \(m\angle C = 95^{\circ}\)

Option 3: \(m\angle A = 43^{\circ}\), \(m\angle B = 32^{\circ}\), \(m\angle C = 95^{\circ}\)

Option 4: \(m\angle A = 53^{\circ}\), \(m\angle B = 95^{\circ}\), \(m\angle C = 32^{\circ}\)

Wait, let's re - calculate the opposite sides. The side opposite \(\angle A\) is \(BC = 8\), opposite \(\angle B\) is \(AC = 12\), opposite \(\angle C\) is \(AB = 15\). So the largest side is \(AB = 15\), so the largest angle is \(\angle C\) (opposite \(AB\)), so \(\angle C = 95^{\circ}\). The second - largest side is \(AC = 12\), so the second - largest angle is \(\angle B\) (opposite \(AC\)), so \(\angle B = 53^{\circ}\). The smallest side is \(BC = 8\), so the smallest angle is \(\angle A\) (opposite \(BC\)), so \(\angle A=32^{\circ}\). Wait, but option 2 is \(m\angle A = 32^{\circ}\), \(m\angle B = 53^{\circ}\), \(m\angle C = 95^{\circ}\), which matches the order of angles (smallest \(\angle A\), middle \(\angle B\), largest \(\angle C\)) based on the side lengths (smallest \(BC = 8\) opposite \(\angle A\), middle \(AC = 12\) opposite \(\angle B\), largest \(AB = 15\) opposite \(\angle C\)).

Wait, maybe I made a mistake in the initial side - angle correspondence. Let's list:

- Side \(BC = 8\): opposite \(\angle A\)
- Side \(AC = 12\): opposite \(\angle B\)
- Side \(AB = 15\): opposite \(\angle C\)

Since \(8<12<15\), then the angles opposite them should satisfy \(m\angle A<m\angle B<m\angle C\). The given angles are \(32^{\circ}\), \(53^{\circ}\), \(95^{\circ}\), and \(32^{\circ}<53^{\circ}<95^{\circ}\). So \(m\angle A = 32^{\circ}\), \(m\angle B = 53^{\circ}\), \(m\angle C = 95^{\circ}\), which is option 2.

Answer:

\(m\angle A = 32^{\circ}, m\angle B = 53^{\circ}, m\angle C = 95^{\circ}\) (corresponding to the option \(m\angle A = 32^{\circ}, m\angle B = 53^{\circ}, m\angle C = 95^{\circ}\))