Question

1. which lines or segments are parallel? justify your answer.

a)

Explanation:

Step1: Identify Angles and Lines

Looking at the diagram, lines \( PS \) and \( QT \) are transversals to the horizontal line (let's call it \( PQ \)) and the base \( ST \). The marked angles at \( P \) and \( Q \) (alternate interior or corresponding?) and also, the angles formed by the diagonal \( PT \) might indicate parallelism. Wait, actually, the vertical segments \( PS \) and \( QT \) – no, wait, the horizontal line \( PQ \) and \( ST \)? Wait, the angles at \( P \) and \( Q \) with the transversal: Wait, the diagram shows \( PQ \) as a horizontal line, \( S \) and \( T \) below, with \( PS \) and \( QT \) as vertical (or same slope) segments? Wait, no, the key is the alternate interior angles or corresponding angles. Wait, the angles at \( P \) (between \( PQ \) and \( PS \)) and at \( T \) (maybe? No, wait, the angles marked at \( P \) and \( Q \) are equal? Wait, the diagram has \( PQ \) horizontal, \( PS \) and \( QT \) as two segments going down to \( S \) and \( T \), and \( ST \) horizontal? Wait, actually, the lines \( PS \) and \( QT \) – wait, no, the correct parallel lines: \( PS \parallel QT \) because the corresponding angles (the angles between \( PQ \) and \( PS \), and \( PQ \) and \( QT \)) are equal (marked as congruent), so by the Corresponding Angles Postulate, \( PS \parallel QT \). Also, \( PQ \parallel ST \) because \( PS \) and \( QT \) are parallel and \( PQST \) looks like a parallelogram? Wait, no, let's re-examine.

Wait, the diagram: \( P \) and \( Q \) are on a horizontal line, \( S \) and \( T \) are below, with \( PS \) connecting \( P \) to \( S \), \( QT \) connecting \( Q \) to \( T \), and \( ST \) connecting \( S \) to \( T \), and \( PT \) is a diagonal. The angles at \( P \) (between \( PQ \) and \( PS \)) and at \( Q \) (between \( PQ \) and \( QT \)) – wait, no, the angles marked are at \( P \) (left of \( PS \)) and at \( Q \) (right of \( QT \))? Wait, maybe the alternate interior angles for \( PQ \) and \( ST \) with transversal \( PS \) or \( QT \). Wait, actually, the correct parallel lines are \( PS \parallel QT \) (because the corresponding angles formed by transversal \( PQ \) are equal) and \( PQ \parallel ST \) (because \( PS \) and \( QT \) are parallel and \( PS \) and \( QT \) are transversals, or because the angles for \( PQ \) and \( ST \) with transversal \( PS \) are equal). Wait, let's clarify:

1. \( PS \parallel QT \): The angles formed by transversal \( PQ \) with \( PS \) and \( QT \) are equal (corresponding angles), so by Corresponding Angles Theorem, \( PS \parallel QT \).

2. \( PQ \parallel ST \): Since \( PS \parallel QT \) and \( PS \) and \( QT \) are both connected to \( S \) and \( T \), the quadrilateral \( PQTS \) has \( PS \parallel QT \) and if \( PQ \parallel ST \), then it's a parallelogram. Alternatively, the angles for \( PQ \) and \( ST \) with transversal \( PS \) – the angle at \( P \) (between \( PQ \) and \( PS \)) and the angle at \( S \) (between \( PS \) and \( ST \)) would be equal (alternate interior angles) if \( PQ \parallel ST \).

But the main parallel lines here are \( PS \parallel QT \) (by Corresponding Angles) and \( PQ \parallel ST \) (by either Corresponding Angles or because \( PS \parallel QT \) and \( PQST \) is a parallelogram).

Wait, let's focus on the marked angles: the angles at \( P \) (between \( PQ \) and \( PS \)) and at \( Q \) (between \( PQ \) and \( QT \)) – if those are equal, then \( PS \parallel QT \) (Corresponding Angles Postulate: if corresponding angles are equal, lines are parallel). Then, since \( PS \) and \( QT \) are parallel, and \( PQ \) and \( ST \) are the other two sides, \( PQ \parallel ST \) (because in a quadrilateral with one pair of parallel sides and equal angles, it's a parallelogram, so both pairs are parallel).

So, the parallel lines/segments are \( PS \parallel QT \) (by Corresponding Angles) and \( PQ \parallel ST \) (by either Corresponding Angles or Parallelogram Properties).

Step1: Analyze Angles for \( PS \parallel QT \)

The angles formed by transversal \( PQ \) with \( PS \) and \( QT \) are congruent (marked as equal). By the Corresponding Angles Postulate, if corresponding angles are equal, the lines are parallel. Thus, \( PS \parallel QT \).

Step2: Analyze Angles for \( PQ \parallel ST \)

Since \( PS \parallel QT \), and \( PS \) and \( QT \) are transversals for \( PQ \) and \( ST \), the alternate interior angles (e.g., \( \angle P S T \) and \( \angle S P Q \)) are equal (because \( PS \parallel QT \) and \( PQ \) is a transversal, but also, the quadrilateral \( PQTS \) has \( PS \parallel QT \) and \( PQ \) and \( ST \) as the other sides, so by the definition of a parallelogram (or by Alternate Interior Angles), \( PQ \parallel ST \).

Answer:

The parallel lines/segments are \( \boldsymbol{PS \parallel QT} \) (by Corresponding Angles Postulate, as the corresponding angles formed by transversal \( PQ \) are congruent) and \( \boldsymbol{PQ \parallel ST} \) (since \( PS \parallel QT \) and \( PQST \) forms a parallelogram, or by Alternate Interior Angles with transversal \( PS \) or \( QT \)).