Question

what are the angle measures of triangle vuw?
$m\angle v = 30^{\circ}$,
$m\angle u = 60^{\circ}$,
$m\angle w = 90^{\circ}$
$m\angle v = 30^{\circ}$,
$m\angle u = 90^{\circ}$,
$m\angle w = 60^{\circ}$
$m\angle v = 60^{\circ}$,
$m\angle u = 90^{\circ}$,
$m\angle w = 30^{\circ}$
$m\angle v = 90^{\circ}$,
$m\angle u = 60^{\circ}$,
$m\angle w = 30^{\circ}$

Explanation:

Step1: Check Pythagorean theorem

Check if \(3^{2}+(3\sqrt{3})^{2}=6^{2}\).
\(3^{2}=9\), \((3\sqrt{3})^{2}=9\times3 = 27\), \(6^{2}=36\).
\(9 + 27=36\), so it is a right - triangle. The side opposite the right - angle is the longest side. So \(\angle V\) is not the right - angle.

Step2: Use sine formula

\(\sin W=\frac{3}{6}=\frac{1}{2}\).
Since \(\sin30^{\circ}=\frac{1}{2}\), so \(m\angle W = 30^{\circ}\).

Step3: Find the third angle

Since the sum of angles in a triangle is \(180^{\circ}\), and one angle is \(90^{\circ}\), another is \(30^{\circ}\). Let \(\angle U\) be the right - angle (\(90^{\circ}\)). Then \(m\angle V=180-(90 + 30)=60^{\circ}\).

Answer:

\(m\angle V = 60^{\circ}\), \(m\angle U = 90^{\circ}\), \(m\angle W = 30^{\circ}\) (the option \(m\angle V = 60^{\circ}\), \(m\angle U = 90^{\circ}\), \(m\angle W = 30^{\circ}\) is correct)