Question

type of angles are supplementary?
wing proof:
8 = 180°

given	reasoning
( mangle 1 cong mangle 5 )	corresponding angle postulate
( mangle 5 + mangle 8 = 180^circ )	linear pair
( mangle 1 + mangle 8 = 180^circ )	substitution pr

Explanation:

Step1: Analyze the Proof Structure

The proof involves parallel lines (\(A \parallel B\)) and angle relationships. We check each step's reasoning. The step with \(m\angle5 \cong m\angle8\) has an incorrect reason. A linear pair of angles are supplementary (sum to \(180^\circ\)), but \(\angle5\) and \(\angle8\) – if they are vertical angles or something else? Wait, no, for parallel lines, consecutive interior angles? Wait, no, the error: the reason for \(m\angle5 \cong m\angle8\) – actually, \(\angle5\) and \(\angle8\) are vertical angles? No, wait, maybe \(\angle5\) and \(\angle8\) are supplementary? Wait, no, the given proof has a mistake in the reasoning for \(m\angle5 \cong m\angle8\). The correct reason for vertical angles is "Vertical Angles Theorem", but if it's a linear pair, linear pair angles are supplementary (sum to \(180^\circ\)), not congruent (unless they are right angles). So the error is in the reasoning for \(m\angle5 \cong m\angle8\); the correct reason should not be "Linear Pair" (since linear pair angles are supplementary, not necessarily congruent) – maybe it's a typo, and the angles are vertical angles, or consecutive interior? Wait, the problem is likely identifying the error in the proof table. Let's re-examine:

• Given \(A \parallel B\), \(m\angle1 \cong m\angle5\) (Corresponding Angles Postulate) – correct.
• Then \(m\angle5 \cong m\angle8\) – the reason given is "Linear Pair" – but linear pair angles are adjacent and form a straight line, so they are supplementary (\(m\angle5 + m\angle8 = 180^\circ\)), not congruent (unless each is \(90^\circ\)). So the error is in the reason for \(m\angle5 \cong m\angle8\); the correct reason should be something else, like "Vertical Angles Theorem" if they are vertical, or maybe a different relationship. But the key is that linear pair angles are supplementary, not congruent, so the reasoning "Linear Pair" for \(m\angle5 \cong m\angle8\) is incorrect.

Step2: Identify the Error

The row with \(m\angle5 \cong m\angle8\) has the wrong reason. Linear pair angles sum to \(180^\circ\) (supplementary), not congruent (equal in measure) unless they are right angles. So the error is in the reasoning for \(m\angle5 \cong m\angle8\); the correct reason should not be "Linear Pair".

Answer:

The error is in the reasoning for \(m\angle5 \cong m\angle8\); the reason "Linear Pair" is incorrect (linear pair angles are supplementary, not necessarily congruent). The correct reason should be related to vertical angles (if they are vertical) or another angle relationship, not linear pair.