Question

in the figure below, lines m and n are cut by transversal t. which of the following is a valid step in the proof that alternate interior angles ∠1 and ∠3 are congruent? ∠1≅∠2 because corresponding angles are congruent ∠1≅∠4 because adjacent angles are congruent m∠1 + m∠4 = 180° because linear pairs are supplementary ∠3≅∠4 because same - side interior angles are congruent

Explanation:

Step1: Recall angle - pair relationships

When two parallel lines \(m\) and \(n\) are cut by a transversal \(t\), we use the properties of angles formed.

Step2: Analyze each option

• Option 1: \(\angle1\cong\angle2\) because corresponding angles are congruent. \(\angle1\) and \(\angle2\) are corresponding angles. If \(m\parallel n\), corresponding angles are congruent. This is a valid step in proving \(\angle1\cong\angle3\) as we can then use the fact that \(\angle2\) and \(\angle3\) are vertical - angles (and vertical angles are congruent) to show \(\angle1\cong\angle3\).
• Option 2: \(\angle1\cong\angle4\) is incorrect because adjacent angles are not generally congruent. Adjacent angles \(\angle1\) and \(\angle4\) form a linear - pair and \(m\angle1 + m\angle4=180^{\circ}\) (linear pairs are supplementary).
• Option 3: \(m\angle1 + m\angle4 = 180^{\circ}\) is a correct property of linear pairs, but it does not directly help in proving \(\angle1\cong\angle3\).
• Option 4: \(\angle3\cong\angle4\) is incorrect because same - side interior angles are supplementary (\(m\angle3 + m\angle4 = 180^{\circ}\)) when \(m\parallel n\), not congruent.

Answer:

\(\angle1\cong\angle2\) because corresponding angles are congruent