Question

examine this figure. which two pieces of information, if true, would help to prove that $\triangle lmp \cong \triangle nmp$ by hl? choose two correct answers.
line $mk$ is the perpendicular bisector of $ln$.
$\overline{ml} \cong \overline{mp}$
point $p$ is the midpoint of $\overline{mk}$.
$\overline{pk} \cong \overline{pk}$
$\overline{ml} \cong \overline{mn}$

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse - Leg) congruence applies to right triangles. So, we need \(\angle LPM\) and \(\angle NPM\) to be right angles, and a leg and hypotenuse congruent.

Step2: Analyze "Line MK is the perpendicular bisector of LN"

If \(MK \perp LN\), then \(\angle LPM = \angle NPM = 90^\circ\), so triangles \(LMP\) and \(NMP\) are right triangles.

Step3: Analyze "\(\overline{ML} \cong \overline{MN}\)"

For HL, hypotenuse \(ML\) (of \(\triangle LMP\)) and hypotenuse \(MN\) (of \(\triangle NMP\)) must be congruent. Also, leg \(MP\) is common to both triangles (\(\overline{MP} \cong \overline{MP}\) by reflexive property). With right angles from the perpendicular bisector, HL is satisfied.

Other options: \(\overline{ML} \cong \overline{MP}\) doesn't relate to hypotenuse/leg for HL. \(P\) as midpoint of \(MK\) or \(\overline{PK} \cong \overline{PK}\) (trivial) don't help HL.

Answer:

• Line MK is the perpendicular bisector of LN.
• \(\overline{ML} \cong \overline{MN}\)