Question

use the given information to prove that ∠6 ≅ ∠1.
you are not allowed to use the transitive property for parallel lines.
(you may use the transitive property for \=\, or \≅\, though.)
given: cb || df
cd || ge
prove: ∠6 ≅ ∠1

Explanation:

Step1: Use parallel line properties

Since \(CB\perp DF\) and \(CD\perp GE\), we know that \(\angle 4 = 90^{\circ}\) (right - angle definition). Also, \(\angle 6\) and \(\angle 4\) are vertical angles. By the vertical - angles theorem, \(\angle 6\cong\angle 4\).

Step2: Use congruence transitivity

\(\angle 1\) and \(\angle 4\) are also vertical angles. So, \(\angle 1\cong\angle 4\) by the vertical - angles theorem. Then, by the transitive property of congruence (\(a\cong b\) and \(b\cong c\) implies \(a\cong c\)), since \(\angle 6\cong\angle 4\) and \(\angle 1\cong\angle 4\), we have \(\angle 6\cong\angle 1\).

Answer:

\(\angle 6\cong\angle 1\) (by vertical - angles theorem and transitive property of congruence)