Question

question 18 (5 points)
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in the given figure, ( mangle tpa = mangle pqc = 135^{circ} ). which one of the following postulates proves the lines ( overleftrightarrow{ab} ) and ( overleftrightarrow{cd} ) are parallel?
a) all of the angles formed by the transversal line ( st ) are congruent.
b) if ( overleftrightarrow{ab} ) and ( overleftrightarrow{cd} ) are congruent lines, then the corresponding angles are congruent.
c) if ( overleftrightarrow{ab} ) and ( overleftrightarrow{cd} ) are parallel lines, then the corresponding angles are congruent.
d) if ( overleftrightarrow{st} ) and ( overleftrightarrow{ab} ) are perpendicular lines, then the corresponding angles are congruent.

Explanation:

• Option A: Not all angles formed by a transversal are congruent. There are supplementary angles as well.
• Option B: Congruent lines do not imply corresponding angles are congruent. Congruent lines refer to length equality, not angle - related properties for parallelism.
• Option C: The converse of the corresponding angles postulate states that if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. Here, \(\angle TPA\) and \(\angle PQC\) are corresponding angles and they are congruent (\(m\angle TPA=m\angle PQC = 135^{\circ}\)), so this postulate can be used to prove \(AB\parallel CD\).
• Option D: Perpendicularity of \(ST\) and \(AB\) (which is not the case here as the angles are \(135^{\circ}\)) is not relevant to the given angle - based parallelism proof.

Answer:

C. If \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) are parallel lines, then the corresponding angles are congruent.