Question

1. 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

Let's 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 on the grid).
For \(EF\): If \(E=(0,8)\) and \(F=(0,2)\), then \(EF=\vert8 - 2\vert=6\)
For \(FD\): If \(F=(0,2)\) and \(D=(3,5)\), then \(FD=\sqrt{(3 - 0)^2+(5 - 2)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\approx4.2\)
For \(ED\): If \(E=(0,8)\) and \(D=(3,5)\), then \(ED=\sqrt{(3 - 0)^2+(5 - 8)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\approx4.2\)

Step2: Use trigonometric ratios

Since \(\triangle EDF\) is a right - triangle at \(D\) (\(\angle EDF = 90^{\circ}\)) and \(ED = FD\), \(\angle DEF=\angle EFD\)
We know that \(\tan\angle EFD=\frac{ED}{FD}=1\) (because \(ED = FD\))
\(\angle EFD=\arctan(1)\) and \(\angle DEF=\arctan(1)\)

Answer:

\(\angle DEF = 45^{\circ}\), \(\angle EFD=45^{\circ}\)