Question

given: $overline{kl}paralleloverline{np}$, $overline{kl}=overline{pn}$ prove: $\triangle kmlcong\triangle pmn$ statements reasons $overline{kl}paralleloverline{np}$ given $overline{kl}congoverline{pn}$ given $angle1congangle4$ $angle2congangle3$ $\triangle klmcong\triangle pnm$

Explanation:

Step1: Identify angle - parallel line relationship

Since $KL\parallel NP$, $\angle1$ and $\angle4$ are alternate interior angles and thus congruent.

Step2: Identify vertical angle relationship

$\angle2$ and $\angle3$ are vertical angles, so they are congruent.

Step3: Apply congruence criterion

With a pair of congruent sides ($KL\cong PN$) and two pairs of congruent angles ($\angle1\cong\angle4$, $\angle2\cong\angle3$), by the ASA congruence criterion, $\triangle KLM\cong\triangle PNM$.

Answer:

• $\angle1\cong\angle4$: Reason - Alternate interior angles are congruent (since $KL\parallel NP$).
• $\angle2\cong\angle3$: Reason - Vertical angles are congruent.
• $\triangle KLM\cong\triangle PNM$: Reason - Angle - Side - Angle (ASA) congruence criterion (we have $\angle1\cong\angle4$, $KL\cong PN$, $\angle2\cong\angle3$).