Question

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

Explanation:

Step1: Check for Pythagorean - triple relationship

In a right - triangle, if the sides are \(a\), \(b\), and \(c\) (\(c\) is the hypotenuse), then \(a^{2}+b^{2}=c^{2}\). Here, \(a = 3\), \(b = 3\sqrt{3}\), and \(c = 6\). Calculate \(a^{2}+b^{2}\): \(3^{2}+(3\sqrt{3})^{2}=9 + 27=36\), and \(c^{2}=6^{2}=36\). So, it is a right - triangle.

Step2: Use trigonometric ratios to find angles

Let's use the sine ratio. \(\sin V=\frac{3\sqrt{3}}{6}=\frac{\sqrt{3}}{2}\), so \(m\angle V = 60^{\circ}\). Since it is a right - triangle, if one non - right angle is \(60^{\circ}\), the other non - right angle \(m\angle W=30^{\circ}\) and \(m\angle U = 90^{\circ}\).

Answer:

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