Question

what are the angle measures of triangle vuw?$mangle v = 30^circ$, $mangle u = 60^circ$, $mangle w = 90^circLXB0mangle v = 30^circ$, $mangle u = 90^circ$, $mangle w = 60^circ$$mangle v = 60^circ$, $mangle u = 90^circ$, $mangle w = 30^circ$(the triangle has side lengths: $wu=3sqrt{3}$ cm, $wv=6$ cm, $uv=3$ cm)

Explanation:

Step1: Check right triangle status

First, verify if $\triangle VUW$ is a right triangle using the Pythagorean theorem. Let $a=3$, $b=3\sqrt{3}$, $c=6$.
Check $a^2 + b^2 = c^2$:

$$\begin{align*} 3^2 + (3\sqrt{3})^2 &= 9 + 27 \\ &= 36 \\ 6^2 &= 36 \end{align*}$$

So $\triangle VUW$ is right-angled, with the right angle opposite the longest side ($6$ cm), which is $\angle U$. Thus $m\angle U = 90^\circ$.

Step2: Find $\angle W$ via sine

Use $\sin(\angle W) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{6} = \frac{1}{2}$.
Since $\sin(30^\circ) = \frac{1}{2}$, $m\angle W = 30^\circ$.

Step3: Calculate remaining angle

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

Answer:

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