To determine the reasoning behind each step in Huy's proof, let's analyze the given statements and their corresponding justifications:
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Statement: \( \text{m}\angle A + \text{m}\angle C = 180^\circ \)
Reason: Linear pairs are supplementary. -
Statement: \( \text{m}\angle A = \text{m}\angle E \)
Reason: This step involves some form of congruence, perhaps utilizing the fact that angles formed by parallel lines and a transversal have certain congruences, but it's not clearly stated in your options. -
Statement: \( \text{m}\angle E + \text{m}\angle C = 180^\circ \)
Reason: Substitution. This is just replacing \( \text{m}\angle A \) with \( \text{m}\angle E \) based on their equality. -
Statement: \( \angle E \) and \( \angle C \) are supplementary.
Reason: Definition of supplementary angles (angles that add up to 180°).
Considering this breakdown, the possible justifications that fit into the proof’s logical flow would be:
- For step 1, if angles A and C are stated to be a linear pair, then "Linear pairs are supplementary" is correct.
- For step 2, assuming \( m\angle A = m\angle E \) corresponds to a property of congruence (perhaps derived from parallel lines or corresponding angles).
- For step 3, the use of \( m\angle A = m\angle E \) to substitute and show \( m\angle E + m\angle C = 180^\circ \) is justified by the Transitive Property of Equality.
- For step 4, the conclusion about angles E and C being supplementary directly comes from the definition of supplementary angles.
Based on this, the correct answers are:
- For step 1: Linear pairs are supplementary.
- For step 2: Corresponding Angles Postulate (if that’s how \( \angle A \) and \( \angle E \) are proven equal).
- For step 3: Transitive Property of Equality.
- For step 4: Definition of supplementary angles.
If you are asked for a specific reasoning behind step arrangements or further clarification, it could be clearer depending on the context given by the options. The main points of clarity relate to recognizing that \( \angle A \) and \( \angle E \) can be shown as equal by properties related to parallel lines, thereby establishing the basis for proving angles E and C as supplementary based on equality and the definition of supplementary angles.
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