Question

an incomplete proof is shown. given: ∠cpn and ∠plh are supplementary. prove: ck || fh. 1. ∠cpn and ∠plh are supplementary 1. given 2. blank 2. definition of supplementary angles 3. ∠cpn ≅ ∠lfk 3. vertical angles theorem 4. m∠cps = m∠lfk 4. definition of congruence 5. blank 5. substitution property of equality 6. blank 6. definition of supplementary angles 7. ck || fh 7. blank

Explanation:

Step1: Apply definition of supplementary angles

$m\angle CPN + m\angle PLH=180^{\circ}$

Step2: Use vertical - angles theorem

Since $\angle CPN\cong\angle LFK$, then $m\angle CPN = m\angle LFK$

Step3: Substitute

Substitute $m\angle CPN$ with $m\angle LFK$ in $m\angle CPN + m\angle PLH = 180^{\circ}$, we get $m\angle LFK+m\angle PLH = 180^{\circ}$

Step4: Apply definition of supplementary angles

$\angle LFK$ and $\angle PLH$ are supplementary

Step5: Use parallel - line theorem

If two same - side interior angles are supplementary, then the two lines are parallel. So, $\overleftrightarrow{CK}\parallel\overleftrightarrow{FH}$

Answer:

1. $m\angle CPN + m\angle PLH = 180^{\circ}$
2. $m\angle LFK+m\angle PLH = 180^{\circ}$
3. $\angle LFK$ and $\angle PLH$ are supplementary
4. If two same - side interior angles are supplementary, then the two lines are parallel.