Question

brooke wants to prove that in the figure shown, \\( \triangle pqt \\) is congruent to \\( \triangle rqs \\) by the side - angle - side postulate.
in brookes proof, which statement would give the justification to show the included angles are congruent?
right angles are congruent.
vertical angles are congruent.
alternate interior angles are congruent.
alternate exterior angles are congruent.

Explanation:

• Vertical angles: Vertical angles are formed when two lines intersect. In the given figure, \(\angle PQT\) and \(\angle RQS\) are vertical angles.
• Right angles: There is no indication in the figure that the angles are right angles (\(90^{\circ}\)).
• Alternate - interior angles: Alternate - interior angles are formed when a transversal intersects two parallel lines. There is no information about parallel lines in the problem description related to these angles.
• Alternate - exterior angles: Similarly, there is no context of parallel lines and a transversal for alternate - exterior angles in relation to \(\angle PQT\) and \(\angle RQS\).

Since \(\angle PQT\) and \(\angle RQS\) are vertical angles, and by the vertical - angles theorem, vertical angles are congruent. This congruent pair of angles is the included angle for the Side - Angle - Side (SAS) postulate for \(\triangle PQT\) and \(\triangle RQS\) (as we can assume from the side - marking in the figure that \(PT = RS\) and \(QT=QS\)).

Answer:

Vertical angles are congruent.