Question

prove:
\\( \overline { d f } = \overline { e f } \\)
helpful hint: start by identifying the given first.

1. \\( \angle 3 = \angle 4 \\)

\\( \angle 1 \\) and \\( \angle 3 \\) are supplementary
2.
\\( \angle 2 \\) and \\( \angle 4 \\) are supplementary

1. \\( \angle 1 = \angle 2 \\)
2. \\( \overline { d f } = \overline { e f } \\)

exterior sides are in opposite rays
given
if the two angles of a triangle are
equal, then the sides opposite those
angles are also equal
two angles supplementary to
equal angles are equal to each other

Explanation:

Step1: Identify the given

$\angle3=\angle4$ is given.

Step2: Use supplementary angles property

Since $\angle1$ and $\angle3$ are supplementary ($\angle1+\angle3 = 180^{\circ}$), and $\angle2$ and $\angle4$ are supplementary ($\angle2+\angle4=180^{\circ}$), and $\angle3=\angle4$, then by the property that two angles supplementary to equal angles are equal to each other, we have $\angle1=\angle2$.

Step3: Apply triangle - side property

In $\triangle DEF$, if $\angle1=\angle2$ (angles of the triangle), then by the property that if the two angles of a triangle are equal, then the sides opposite those angles are also equal. So, $\overline{DF}=\overline{EF}$.

Answer:

1. $\angle3 = \angle4$: Given
2. $\angle1$ and $\angle3$ are supplementary, $\angle2$ and $\angle4$ are supplementary: Exterior Sides are in Opposite Rays
3. $\angle1=\angle2$: Two Angles Supplementary to Equal Angles are Equal to Each Other
4. $\overline{DF}=\overline{EF}$: If the Two Angles of a Triangle are Equal, then the Sides Opposite those Angles are also Equal