Question

what are the angle measures of triangle vuw? o m∠v = 30°, m∠u = 60°, m∠w = 90° o m∠v = 90°, m∠u = 60°, m∠w = 30° o m∠v = 30°, m∠u = 90°, m∠w = 60° o m∠v = 60°, m∠u = 90°, m∠w = 30°

Explanation:

Step1: Check Pythagorean - theorem

We know that in a right - triangle, if the sides are \(a\), \(b\), and \(c\) (\(c\) is the hypotenuse), then \(a^{2}+b^{2}=c^{2}\). Here, \(3^{2}+(3\sqrt{3})^{2}=9 + 27=36=6^{2}\). So, triangle \(VUW\) is a right - triangle, and the hypotenuse is \(UW = 6\) cm.

Step2: Use trigonometric ratios

Let's use the sine ratio. \(\sin V=\frac{UW}{VW}\). \(\sin V=\frac{3\sqrt{3}}{6}=\frac{\sqrt{3}}{2}\), so \(m\angle V = 60^{\circ}\).

Step3: Find the third angle

Since the sum of the interior angles of a triangle is \(180^{\circ}\), and one angle is \(90^{\circ}\) (because it's a right - triangle) and \(m\angle V = 60^{\circ}\), then \(m\angle W=180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}\), and \(m\angle U = 90^{\circ}\).

Answer:

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