Question

1. given: ( cparallel d ), ( angle1congangle2 ) prove: ( aparallel b )

Explanation:

Step1: Use the property of parallel lines

Since \(c\parallel d\), by the property of parallel lines (alternate - interior angles), \(\angle1=\angle3\) (corresponding angles).

Step2: Substitute the given angle equality

Given \(\angle1\cong\angle2\), from Step1 \(\angle1 = \angle3\), so \(\angle2=\angle3\) (by substitution).

Step3: Apply the converse of parallel - line theorem

If the corresponding angles are equal, then the lines are parallel. Since \(\angle2\) and \(\angle3\) are corresponding angles, \(a\parallel b\) (by the converse of the corresponding - angles postulate).

Answer:

Statement	Reason
2. \(\angle1\cong\angle2\)	Given
3. \(\angle1=\angle3\)	Corresponding angles (because \(c\parallel d\))
4. \(\angle2=\angle3\)	Substitution (from 2 and 3)
5. \(a\parallel b\)	Converse of corresponding - angles postulate (because \(\angle2\) and \(\angle3\) are corresponding angles)