Question

given: ∠cpn and ∠plh are supplementary. prove: (overleftrightarrow{ck}paralleloverleftrightarrow{fh}). 1. ∠cpn and ∠plh are supplementary. 1. given. 2. (mangle cpn + mangle hlp=180^{circ}). 2. definition of supplementary angles. 3. ∠cpn≅∠lpk. 3. vertical - angles theorem. 4. (mangle cpn = mangle lpk). 4. definition of congruence. 5. (mangle lpk + mangle hlp = 180^{circ}). 5. substitution property of equality. 6. ∠lpk and ∠hlp are supplementary. 6. definition of supplementary angles.

Explanation:

Step1: Recall supplementary - angle def

By the definition of supplementary angles, if two angles are supplementary, the sum of their measures is 180 degrees. So, if $\angle CPN$ and $\angle PLH$ are supplementary, then $m\angle CPN + m\angle PLH=180^{\circ}$.

Step2: Use vertical - angles theorem

Vertical angles are congruent. $\angle CPN$ and $\angle LPK$ are vertical angles, so $\angle CPN\cong\angle LPK$ by the vertical - angles theorem.

Step3: Apply congruence definition

Since $\angle CPN\cong\angle LPK$, by the definition of congruence, $m\angle CPN = m\angle LPK$.

Step4: Substitute

Substitute $m\angle CPN$ with $m\angle LPK$ in the equation $m\angle CPN + m\angle PLH = 180^{\circ}$. We get $m\angle LPK+m\angle PLH = 180^{\circ}$.

Step5: Use supplementary - angle def again

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

Step6: Use converse of same - side interior angles postulate

If two lines are cut by a transversal and a pair of same - side interior angles are supplementary, then the two lines are parallel. $\angle LPK$ and $\angle PLH$ are same - side interior angles formed by lines $\overleftrightarrow{CK}$ and $\overleftrightarrow{FH}$ with transversal $\overleftrightarrow{MN}$. Since $\angle LPK$ and $\angle PLH$ are supplementary, $\overleftrightarrow{CK}\parallel\overleftrightarrow{FH}$.

Answer:

The proof is completed as above to show that $\overleftrightarrow{CK}\parallel\overleftrightarrow{FH}$ based on the given information and geometric properties.