Question

1. ∠cpn and ∠plh are supplementary. given
2. m∠cpn + m∠hlp = 180°. definition of supplementary angles
3. ∠cpn ≅ ∠lpk. vertical angles theorem
4. m∠cpn = m∠lpk. definition of congruence
5. m∠lpk + m∠hlp = 180°. substitution property of equality
6. ∠lpk and ∠hlp are supplementary. definition of supplementary angles
7. ck∥ph. if two lines are intersected by a transversal so that consecutive interior angles are supplementary, then the lines are parallel

Explanation:

Step1: Given supplementary angles

$\angle CPN$ and $\angle PLH$ are supplementary, so $m\angle CPN + m\angle HLP=180^{\circ}$ by the definition of supplementary angles.

Step2: Use vertical - angles theorem

$\angle CPN\cong\angle LPK$ by the vertical - angles theorem, and by the definition of congruence, $m\angle CPN = m\angle LPK$.

Step3: Apply substitution

Substitute $m\angle CPN$ with $m\angle LPK$ in $m\angle CPN + m\angle HLP = 180^{\circ}$, getting $m\angle LPK+m\angle HLP = 180^{\circ}$.

Step4: Define supplementary angles

Since $m\angle LPK+m\angle HLP = 180^{\circ}$, $\angle LPK$ and $\angle HLP$ are supplementary by the definition of supplementary angles.

Step5: Use parallel - lines theorem

Given that $\angle LPK$ and $\angle HLP$ are consecutive interior angles formed by lines $\overleftrightarrow{CK}$ and $\overleftrightarrow{PH}$ with a transversal, and they are supplementary, then $\overleftrightarrow{CK}\parallel\overleftrightarrow{PH}$ by the theorem: if two lines are intersected by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

Answer:

The proof is complete as shown above to prove that $\overleftrightarrow{CK}\parallel\overleftrightarrow{PH}$.