Question

1. polygon hpyn is shown.

a. mark the diagram to show the following given information.
given: \\( \overline { h p } \cong \overline { y n } \\) and \\( \overline { h p } \parallel \overline { y n } \\)
d. complete the two - column proof.

Explanation:

Step1: Fill in the reason for statement 2

Since it is given in the problem, the reason for $\overline{HP}\parallel\overline{YN}$ is "Given".

Step2: Identify the alternate - interior angles

When two parallel lines ($\overline{HP}$ and $\overline{YN}$) are cut by a transversal ($\overline{PN}$), $\angle HPN\cong\angle YNP$. The reason is "Alternate - interior angles theorem".

Step3: Use the side - angle - side (SAS) congruence criterion

We have $\overline{HP}\cong\overline{YN}$ (given), $\angle HPN\cong\angle YNP$ (from step 2), and $\overline{PN}\cong\overline{NP}$ (reflexive property of congruence).

Answer:

Statement	Reason
2. $\overline{HP}\parallel\overline{YN}$	2. Given
3. $\angle HPN\cong\angle YNP$	3. Alternate - interior angles theorem
4. $\overline{PN}\cong\overline{NP}$	4. Reflexive property of congruence
5. $\triangle PHN\cong\triangle NYP$	5. SAS (Side - Angle - Side) congruence criterion