Question

1. write a proof.

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

Explanation:

Step1: Determine the measure of ∠2

Since \( \angle1\cong\angle2\) and \( \angle1 + \angle2=180^{\circ}\) (linear - pair postulate), then \(2\angle2 = 180^{\circ}\), so \( \angle2=90^{\circ}\).

Step2: Use the property of parallel lines

Given \(c\parallel d\) and \(b\perp d\). By the property that if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. So \(b\perp c\).

Step3: Conclude \(a\parallel b\)

Since \(a\perp c\) (\(\angle2 = 90^{\circ}\), so \(a\) is perpendicular to \(c\)) and \(b\perp c\), by the theorem that if two lines are perpendicular to the same line, then the two lines are parallel. So \(a\parallel b\).

Answer:

By showing \(a\perp c\) (because \(\angle2 = 90^{\circ}\)) and \(b\perp c\) (using the property of parallel lines \(c\parallel d\) and \(b\perp d\)), we conclude \(a\parallel b\) (by the theorem of two lines perpendicular to the same line are parallel).