Question

use the law of sines to solve for b.
enter \\(m\angle c\\).
\\(\frac{\sin(?)}{\box} = \frac{\sin(\box)}{b}\\)
law of sines: \\(\frac{\sin a}{a} = \frac{\sin b}{b} = \frac{\sin c}{c}\\)

Explanation:

Find the measure of angle C

Using the Triangle Angle Sum knowledge point

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Set up the Law of Sines equation

We need to fill in the blanks for the equation:

$$ \frac{\sin([?])}{\text{blank}} = \frac{\sin(\text{blank})}{b} $$

According to the Law of Sines:

$$ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $$

Since the right side of the equation has \(b\) in the denominator, its numerator must contain the angle opposite to side \(b\), which is angle \(B\).
Thus, the right numerator is \(\sin(69^\circ)\).

The left side must use the known side \(c = 6\) and its opposite angle \(C = 62^\circ\).
Thus, the left side is:

$$ \frac{\sin(62^\circ)}{6} $$

Comparing this to the template, the green box with the question mark \([?]\) represents the angle in the numerator of the left ratio, which is \(62^\circ\).
The prompt specifically asks to "Enter \(m\angle C\)" which corresponds to the green box \([?]\).
Therefore, the value to enter is \(62\).

Answer:

Enter \(m\angle C\).

$$\frac{\sin([\text{ 62 }])}{6} = \frac{\sin(69)}{b}$$