Question

what are the angle measures of triangle vuw?

• ( mangle v = 30^circ, mangle u = 60^circ, mangle w = 90^circ )
• ( mangle v = 90^circ, mangle u = 60^circ, mangle w = 30^circ )
• ( mangle v = 30^circ, mangle u = 90^circ, mangle w = 60^circ )
• ( mangle v = 60^circ, mangle u = 90^circ, mangle w = 30^circ )

Explanation:

Step1: Verify right triangle

Check if sides satisfy Pythagorean theorem:
$$(3)^2 + (3\sqrt{3})^2 = 9 + 27 = 36 = 6^2$$

Step2: Identify right angle

Right angle is opposite hypotenuse (6 cm), so $\angle U = 90^\circ$.

Step3: Find $\angle W$

Use sine: $\sin W = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{6} = \frac{1}{2}$, so $m\angle W = 30^\circ$.

Step4: Calculate $\angle V$

Sum of angles in triangle is $180^\circ$:
$$m\angle V = 180^\circ - 90^\circ - 30^\circ = 60^\circ$$

Answer:

$m\angle V = 60^\circ, m\angle U = 90^\circ, m\angle W = 30^\circ$