Question

determining missing components of right triangles
use the information given in the diagram of △ghi to find the missing measurements. round all answers to the nearest tenth.
m∠g =
hi =
gh =
(diagram shows right triangle ghi with right angle at h, ∠i is 39°, side gi is 30)

Explanation:

Step1: Calculate $m\angle G$

The sum of angles in a triangle is $180^\circ$. For right triangle $\triangle GHI$, $\angle H=90^\circ$, $\angle I=39^\circ$.
$m\angle G = 180^\circ - 90^\circ - 39^\circ = 51^\circ$

Step2: Calculate length $HI$

Use cosine of $\angle I$: $\cos(\angle I)=\frac{HI}{GI}$. $GI=30$, $\angle I=39^\circ$.
$HI = GI \times \cos(39^\circ) = 30 \times \cos(39^\circ) \approx 30 \times 0.7771 = 23.3$

Step3: Calculate length $GH$

Use sine of $\angle I$: $\sin(\angle I)=\frac{GH}{GI}$. $GI=30$, $\angle I=39^\circ$.
$GH = GI \times \sin(39^\circ) = 30 \times \sin(39^\circ) \approx 30 \times 0.6293 = 18.9$

Answer:

$m\angle G = 51^\circ$
$HI = 23.3$
$GH = 18.9$