Question

use the given information to prove that \\( \overline{bc} \parallel \overline{fd} \\).
you are not allowed to use the transitive property for parallel lines.
(you may use the transitive property for \= \ or \\cong\, though.)

given: \\( \overline{ba} \perp \overline{ce} \\)
\\( \angle 1 \cong \angle 6 \\)
prove: \\( \overline{bc} \parallel \overline{fd} \\)

step | statement | reason
1 | | reason?

Explanation:

Step1: Identify right angles

Since $\overline{BA} \perp \overline{CE}$, $\angle 1 = 90^\circ$ and $\angle 2 = 90^\circ$ (definition of perpendicular lines).

Step2: Use given congruent angles

Given $\angle 1 \cong \angle 6$, so $\angle 6 = 90^\circ$ (definition of congruent angles).

Step3: Relate angles for parallelism

$\angle 2 = 90^\circ$ and $\angle 6 = 90^\circ$, so $\angle 2 \cong \angle 6$ (transitive property of equality). $\angle 2$ and $\angle 6$ are corresponding angles; if corresponding angles are congruent, lines are parallel.

Answer:

$\overline{BC} \parallel \overline{FD}$ is proven.