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 lines

We are given lines \(AB\) and \(CD\) intersected by a transversal line. Let the intersection point on line \(AB\) be \(P\) and on line \(CD\) be \(Q\).
The given angle measures from the diagram are:

• The interior angle on the left side of the transversal at \(P\) is \(148^\circ\).
• The interior angle on the right side of the transversal at \(Q\) is \(32^\circ\).

Calculate the adjacent supplementary angle

Using the Straight Angle Theorem knowledge point

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Identify consecutive interior angles

We examine the interior angles on the same side of the transversal line \(PQ\):

• The interior angle at \(P\) on the left side is \(\angle APQ = 148^\circ\).
• The interior angle at \(Q\) on the left side is \(\angle CQP = 180^\circ - 32^\circ = 148^\circ\).
• Alternatively, the consecutive interior angles on the left side are \(\angle APQ\) and \(\angle CQP\).
• Let's sum the consecutive interior angles on the left side:

$$ \angle APQ + \angle CQP = 148^\circ + 32^\circ = 180^\circ $$

Apply the parallel line converse theorem

Using the Consecutive Interior Angles Converse knowledge point

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Construct the two-column proof

We organize the steps into a formal statement-reason table.

1. \(\angle APQ = 148^\circ\) and \(\angle CQP = 32^\circ\) | Given
2. \(\angle APQ + \angle CQP = 148^\circ + 32^\circ = 180^\circ\) | Addition of angle measures
3. \(AB \parallel CD\) | Consecutive Interior Angles Converse

Answer:

The formal two-column proof to show that lines \(AB\) and \(CD\) are parallel is as follows:

Statement	Reason
2. \(\angle APQ + \angle CQP = 180^\circ\)	2. Addition Property of Equality
3. \(AB \parallel CD\)	3. Consecutive Interior Angles Converse