Question

find the measures of \\( \angle d e f \\) and \\( \angle e f d \\). round to the nearest tenth, if necessary.

Explanation:

Step1: Calculate the lengths of the sides

Assume the side - length of each square is \(1\).
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (or counting the units in the coordinate - grid).
Let \(E=(0,6)\), \(D=(3,4)\), \(F=(0,1)\)
\(ED=\sqrt{(3 - 0)^2+(4 - 6)^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.6\)
\(FD=\sqrt{(3 - 0)^2+(4 - 1)^2}=\sqrt{9+9}=\sqrt{18}\approx4.2\)
\(EF=\sqrt{(0 - 0)^2+(6 - 1)^2}=5\)

Step2: Use trigonometric ratios

For \(\angle DEF\):
\(\sin\angle DEF=\frac{FD}{EF}\) (since in right - triangle \(EDF\) with right - angle at \(D\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\))
\(\sin\angle DEF=\frac{\sqrt{18}}{5}\approx\frac{4.2}{5} = 0.84\)
\(\angle DEF=\sin^{-1}(0.84)\approx57.1^{\circ}\)

For \(\angle EFD\):
\(\sin\angle EFD=\frac{ED}{EF}\) (since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) in right - triangle \(EDF\))
\(\sin\angle EFD=\frac{\sqrt{13}}{5}\approx\frac{3.6}{5}=0.72\)
\(\angle EFD=\sin^{-1}(0.72)\approx46.1^{\circ}\)

Answer:

\(\angle DEF\approx57.1^{\circ}\) and \(\angle EFD\approx46.1^{\circ}\)