Question

use the information given in the diagram of △ghi to find the missing measurements. round all answers to the nearest tenth. m∠g = \boxed{} hi = \boxed{} gm = \boxed{}

Explanation:

Step1: Find \( m\angle G \)

In a right triangle, the sum of angles is \( 180^\circ \). Given \( \angle M = 90^\circ \) and \( \angle I = 30^\circ \), so \( m\angle G=180^\circ - 90^\circ - 30^\circ = 60^\circ \).

Step2: Find \( \text{IM} \)

In a \( 30^\circ - 60^\circ - 90^\circ \) triangle, the side opposite \( 30^\circ \) (here \( \text{IM} \) opposite \( \angle G = 60^\circ \)? Wait, no, \( \text{OI} = 10 \) is the hypotenuse. Wait, \( \angle I = 30^\circ \), so the side opposite \( 30^\circ \) is \( \text{GM} \), and the side opposite \( 60^\circ \) is \( \text{IM} \), hypotenuse \( \text{OI}=10 \). Wait, maybe \( \text{OI} \) is hypotenuse, so \( \text{GM}=\frac{1}{2}\times\text{OI}=\frac{1}{2}\times10 = 5 \), and \( \text{IM}=\text{GM}\times\sqrt{3}=5\sqrt{3}\approx8.7 \), and \( m\angle G = 60^\circ \). Wait, let's re - check. In right triangle \( \triangle GMI \) (assuming right angle at \( M \)), \( \angle I = 30^\circ \), hypotenuse \( \text{GI}=10 \) (wait, the diagram has \( \text{OI} = 10 \), maybe typo, should be \( \text{GI}=10 \)). Then:

• \( m\angle G=90^\circ - 30^\circ = 60^\circ \) (since in right triangle, acute angles are complementary).
• \( \text{IM}=\text{GI}\times\cos(30^\circ)=10\times\frac{\sqrt{3}}{2}=5\sqrt{3}\approx8.7 \)
• \( \text{GM}=\text{GI}\times\sin(30^\circ)=10\times\frac{1}{2}=5 \)

Answer:

\( m\angle G=\boldsymbol{60^\circ} \), \( \text{IM}\approx\boldsymbol{8.7} \), \( \text{GM}=\boldsymbol{5} \) (assuming \( \text{GI} = 10 \) as the hypotenuse, adjust if the side label is different)