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Angles are created when lines \(a\) and \(b\) are cut by a transversal, \(t\). These angles are labeled in the diagram. Select the claim that is true about lines \(a\) and \(b\) for all cases.

A. If lines \(a\) and \(b\) are cut by a transversal, \(t\), such that certain angle relationships hold, then the lines must be parallel.
B. If lines \(a\) and \(b\) are cut by a transversal, \(t\), such that certain angle relationships hold, then the lines must be parallel.
C. If lines \(a\) and \(b\) are cut by a transversal, \(t\), such that certain angle relationships hold, then the lines must be perpendicular.
D. If lines \(a\) and \(b\) are cut by a transversal, \(t\), such that certain angle relationships hold, then the lines must be perpendicular.

Answer :

The correct claim based on geometric theorems is that if a transversal cuts two lines such that alternate interior angles are congruent, the lines must be parallel. There were no valid claims provided that would guarantee the lines are perpendicular based solely on angle measures.

The relationship between lines and angles formed when a transversal cuts two other lines. According to the information provided and several established geometric theorems, specific angle relationships can determine whether two lines are parallel, perpendicular, or neither.

Theorem 19 states that if two parallel lines are cut by a transversal, the alternate-interior angles are congruent. Conversely, if alternate-interior angles formed by a transversal are congruent, it can be deduced that the lines are parallel. In contrast, no information was provided on a situation where the lines are necessarily perpendicular based on angle measures alone.

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Rewritten by : Barada

When lines a and b are cut by a transversal, t, such that , , , and , then the lines must be parallel '

The Angle-Angle criteria for parallel lines states that if two lines are cut by a transversal and the accompanying angles are congruent, the lines must be parallel.

Lines a and b must be parallel if they are intersected by a transversal, t, such that,,, and this is because when a transversal cuts two lines, the angles on opposite sides of the transversal are supplementary angles, which means they sum up to 180 degrees. This signifies that the angles on the transversal's same side are congruent, or equal. When the angles on the same side of the transversal are equal, the lines are parallel.

This is true in all circumstances because when a transversal cuts two lines, the angles on opposite sides of the transversal are always supplementary, which means they always add up to 180 degrees. The angles on the same side of the transversal are always congruent, or equal. If the angles on the same side of the transversal are equal, the lines are parallel.

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