Consider the triangle. The measures of the angles of the triangle are [tex]32^{\circ}, 53^{\circ}, 95^{\circ}[/tex]. Based on the side lengths, what are the measures of each angle?

A. [tex]m \angle A = 95^{\circ}, m \angle B = 53^{\circ}, m \angle C = 32^{\circ}[/tex]
B. [tex]m \angle A = 32^{\circ}, m \angle B = 53^{\circ}, m \angle C = 95^{\circ}[/tex]
C. [tex]m \angle A = 43^{\circ}, m \angle B = 32^{\circ}, m \angle C = 95^{\circ}[/tex]
D. [tex]m \angle A = 53^{\circ}, m \angle B = 95^{\circ}, m \angle C = 32^{\circ}[/tex]



Answer :

To determine the correct measure of each angle in the given triangle, we need to match the provided options with the given angles. The angles given are [tex]\(32^\circ\)[/tex], [tex]\(53^\circ\)[/tex], and [tex]\(95^\circ\)[/tex].

Let's compare each option with the given angles:

1. Option 1:
[tex]\[ m \angle A = 95^\circ, \quad m \angle B = 53^\circ, \quad m \angle C = 32^\circ \][/tex]
This option matches the given angles.

2. Option 2:
[tex]\[ m \angle A = 32^\circ, \quad m \angle B = 53^\circ, \quad m \angle C = 95^\circ \][/tex]
This option also matches the given angles.

3. Option 3:
[tex]\[ m \angle A = 43^\circ, \quad m \angle B = 32^\circ, \quad m \angle C = 95^\circ \][/tex]
This option does not match the given angles as [tex]\(43^\circ\)[/tex] is not one of the given angles.

4. Option 4:
[tex]\[ m \angle A = 53^\circ, \quad m \angle B = 95^\circ, \quad m \angle C = 32^\circ \][/tex]
This option again matches the given angles.

So, the measures of each angle in the triangle based on the side lengths can be summarized as:
- [tex]\(m \angle A = 95^\circ\)[/tex], [tex]\(m \angle B = 53^\circ\)[/tex], [tex]\(m \angle C = 32^\circ\)[/tex]
- [tex]\(m \angle A = 32^\circ\)[/tex], [tex]\(m \angle B = 53^\circ\)[/tex], [tex]\(m \angle C = 95^\circ\)[/tex]
- [tex]\(m \angle A = 53^\circ\)[/tex], [tex]\(m \angle B = 95^\circ\)[/tex], [tex]\(m \angle C = 32^\circ\)[/tex]

These match the options:

1. [tex]\(m \angle A = 95^\circ, m \angle B = 53^\circ, m \angle C = 32^\circ\)[/tex]
2. [tex]\(m \angle A = 32^\circ, m \angle B = 53^\circ, m \angle C = 95^\circ\)[/tex]
4. [tex]\(m \angle A = 53^\circ, m \angle B = 95^\circ, m \angle C = 32^\circ\)[/tex]

Option 3 is incorrect.

Thus, the correct angle measures consistent with the given angles are options 1, 2, and 4.