Question

1. given: l || m; ∠4≅∠□ prove: ∠8 and ∠9 are supplementary statements reasons

Explanation:

Step1: Identify corresponding - angles

Since \(l\parallel m\), \(\angle4\) and \(\angle8\) are corresponding angles. So, \(\angle4\cong\angle8\) (Corresponding - angles postulate).

Step2: Use the given congruence

We are given that \(\angle4\cong\angle9\).

Step3: Apply the transitive property of congruence

By the transitive property of congruence (if \(a = b\) and \(b = c\), then \(a = c\)), since \(\angle4\cong\angle8\) and \(\angle4\cong\angle9\), we have \(\angle8\cong\angle9\).

Step4: Recall the definition of supplementary angles

We know that \(\angle8\) and \(\angle9\) form a linear - pair. A linear - pair of angles is supplementary, that is, the sum of the measures of two angles in a linear - pair is \(180^{\circ}\). So, \(\angle8\) and \(\angle9\) are supplementary.

Answer:

Statements:

1. \(l\parallel m\) (Given)
2. \(\angle4\cong\angle8\) (Corresponding - angles postulate)
3. \(\angle4\cong\angle9\) (Given)
4. \(\angle8\cong\angle9\) (Transitive property of congruence)
5. \(\angle8\) and \(\angle9\) are supplementary (Definition of linear - pair of angles)

Reasons:

1. Given
2. Corresponding - angles postulate
3. Given
4. Transitive property of congruence
5. Definition of linear - pair of angles