Question

1 list the definitions, postulates, and theorems abelina and madison used in their proof plans. explain your reasoning.
2 use madisons proof plan to complete the flowchart proof of the alternate interior angle theorem by writing the statement for each reason in the boxes.
3 use abelinas proof plan to create a two - column proof.

Explanation:

Step1: Analyze problem requirements

We need to list relevant geometric concepts for proof - plans and complete proof - related tasks.

Step2: For question 1

Definitions, postulates and theorems used could be based on geometric properties like parallel - line properties. The reasoning is that in geometric proofs, we rely on established definitions (e.g., of angles), postulates (e.g., corresponding - angle postulate) and theorems (e.g., vertical - angles theorem) to build logical arguments.

Step3: For question 2

Given: Assume two parallel lines cut by a transversal.
Corresponding Angle Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. Let \(\angle1\) and \(\angle5\) be corresponding angles, so \(\angle1\cong\angle5\).
Vertical angles are congruent: If we have two intersecting lines forming vertical angles, say \(\angle5\) and \(\angle3\) are vertical angles, so \(\angle5\cong\angle3\).
Transitive Property: Since \(\angle1\cong\angle5\) and \(\angle5\cong\angle3\), by the transitive property of congruence, \(\angle1\cong\angle3\) (where \(\angle1\) and \(\angle3\) are alternate - interior angles).

Step4: For question 3

Statements:

1. Two parallel lines \(l\) and \(m\) are cut by a transversal \(t\). (Given)
2. \(\angle A\) and \(\angle B\) are corresponding angles. (Definition of corresponding angles)
3. \(\angle A\cong\angle B\). (Corresponding Angle Postulate)
4. \(\angle B\) and \(\angle C\) are vertical angles. (Definition of vertical angles)
5. \(\angle B\cong\angle C\). (Vertical angles are congruent)
6. \(\angle A\cong\angle C\). (Transitive Property of Congruence)

Reasons:

1. Given
2. Definition of corresponding angles
3. Corresponding Angle Postulate
4. Definition of vertical angles
5. Vertical angles are congruent
6. Transitive Property of Congruence

Answer:

1. Definitions, postulates and theorems used may include definitions of angles, corresponding - angle postulate, vertical - angles theorem etc. Reasoning: Geometric proofs are built on these established facts.
2. Given: Two parallel lines cut by a transversal. Corresponding Angle Postulate: Corresponding angles are congruent. Vertical angles are congruent: For vertical - angle pairs. Transitive Property: To show congruence of alternate - interior angles.
3. Statements: 1. Two parallel lines \(l\) and \(m\) are cut by a transversal \(t\). 2. \(\angle A\) and \(\angle B\) are corresponding angles. 3. \(\angle A\cong\angle B\). 4. \(\angle B\) and \(\angle C\) are vertical angles. 5. \(\angle B\cong\angle C\). 6. \(\angle A\cong\angle C\). Reasons: 1. Given 2. Definition of corresponding angles 3. Corresponding Angle Postulate 4. Definition of vertical angles 5. Vertical angles are congruent 6. Transitive Property of Congruence