Question

1. write a proof.

given \\( \angle 1 \cong \angle 2 \\),
\\( c \parallel d \\), and
\\( b \perp d \\)
prove \\( a \parallel b \\)

Explanation:

Step1: Use the property of vertical angles

Since \(\angle1\cong\angle2\) and \(\angle1\) and \(\angle2\) are adjacent angles forming a linear pair (\(\angle1+\angle2 = 180^{\circ}\), because \(\angle1\cong\angle2\)), we get \(\angle1=\angle2 = 90^{\circ}\), so \(a\perp c\).

Step2: Use the property of parallel lines

Given \(c\parallel d\), if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. Since \(a\perp c\), then \(a\perp d\).

Step3: Use the property of perpendicular lines

Given \(b\perp d\) and \(a\perp d\). In a plane, if two lines are perpendicular to the same line, then the two lines are parallel.

Answer:

Since \(a\perp d\) (from \(a\perp c\) and \(c\parallel d\)) and \(b\perp d\), by the theorem that in a plane, if two lines are perpendicular to the same line, then the two lines are parallel, we have \(a\parallel b\).