Question

find ( mangle g ).
triangle image with right angle at i, ( hi = 2sqrt{37} ), ( gi = 2sqrt{111} ), right angle at i
( mangle g = square^circ )

Explanation:

Step1: Identify trigonometric ratio

In right triangle \( \triangle GHI \) with right angle at \( I \), to find \( m\angle G \), we can use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. For \( \angle G \), the opposite side is \( HI = 2\sqrt{37} \) and the adjacent side is \( GI = 2\sqrt{111} \). So, \( \tan(G)=\frac{HI}{GI} \).

Step2: Simplify the ratio

Substitute the values: \( \tan(G)=\frac{2\sqrt{37}}{2\sqrt{111}} \). The 2s cancel out, and we can simplify \( \frac{\sqrt{37}}{\sqrt{111}} \). Note that \( \sqrt{111}=\sqrt{3\times37} \), so \( \frac{\sqrt{37}}{\sqrt{3\times37}}=\frac{1}{\sqrt{3}} \).

Step3: Find the angle

We know that \( \tan(G)=\frac{1}{\sqrt{3}} \), and from the unit circle or trigonometric values, \( \tan(30^\circ)=\frac{1}{\sqrt{3}} \). So, \( m\angle G = 30^\circ \).

Answer:

\( 30 \)