Question

1. prove that lines ab and cd are parallel.

statement | reason

1. given that ab || cd and \angle 1 = \angle 2, prove ac = ab.

Explanation:

Identify the given angles and transversal

Let the intersection of the transversal line with line \(AB\) be point \(P\), and with line \(CD\) be point \(Q\).
We are given:

• The interior angle on the left side of the transversal at line \(AB\) is \(\angle APQ = 148^\circ\).
• The interior angle on the right side of the transversal at line \(CD\) is \(\angle PQD = 32^\circ\).

Find the consecutive interior angle

Using the Straight Angle Theorem knowledge point

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Check the consecutive interior angles sum

Using the Consecutive Interior Angles Converse knowledge point

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Since the consecutive interior angles do not sum to \(180^\circ\), we must check alternate interior angles.

Analyze alternate interior angles

Using the Alternate Interior Angles knowledge point

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Since the alternate interior angles \(\angle BPQ\) and \(\angle PQD\) are equal (\(32^\circ = 32^\circ\)), the lines \(AB\) and \(CD\) are parallel.

Construct the two-column proof

We organize the statements and reasons systematically:

1. \(\angle APQ = 148^\circ\) (Given)
2. \(\angle BPQ + \angle APQ = 180^\circ\) (Linear Pair Postulate / Straight Angle Theorem)
3. \(\angle BPQ = 32^\circ\) (Subtraction Property of Equality)
4. \(\angle PQD = 32^\circ\) (Given)
5. \(\angle BPQ = \angle PQD\) (Transitive Property of Equality)
6. \(AB \parallel CD\) (Alternate Interior Angles Converse)

Answer:

Two-Column Proof

Statement	Reason
2. \(\angle BPQ + \angle APQ = 180^\circ\)	2. Linear Pair Postulate
3. \(\angle BPQ = 32^\circ\)	3. Subtraction Property of Equality
4. \(\angle PQD = 32^\circ\)	4. Given
5. \(\angle BPQ = \angle PQD\)	5. Transitive Property of Equality
6. \(AB \parallel CD\)	6. Alternate Interior Angles Converse