Question

mark the figure to show the given information. explain how you would prove \\( \angle p \cong \angle q \\) given: \\( \overline{kl} \\) is the perpendicular bisector of \\( \overline{pq} \\) which figure below has the correct marks to show the given information? complete the explanation of how to prove \\( \angle p \cong \angle q \\) below. use to show that \\( \overline{pl} \cong \overline{ql} \\), and use to show that \\( \angle plk \cong \angle qlk \\).

Explanation:

Since \( \overline{KL} \) is the perpendicular bisector of \( \overline{PQ} \), by the definition of a perpendicular bisector, it divides \( \overline{PQ} \) into two equal parts (\( \overline{PL}\cong\overline{QL} \)) and forms right angles (\( \angle PLK\cong\angle QLK = 90^{\circ} \)).

Answer:

Use the definition of a perpendicular bisector to show that \( \overline{PL}\cong\overline{QL} \), and use the definition of perpendicular lines (or the fact that perpendicular lines form congruent right - angles) to show that \( \angle PLK\cong\angle QLK \). Then, since \( \overline{KL}\cong\overline{KL} \) (reflexive property), by the Side - Angle - Side (SAS) congruence criterion, \( \triangle PLK\cong\triangle QLK \). And if two triangles are congruent, their corresponding angles are congruent, so \( \angle P\cong\angle Q \).