Question

given: ∠cpn and ∠plh are supplementary
prove: $overleftrightarrow{ck} \parallel overleftrightarrow{fh}$

statement	reason
2. $square$	2. definition of supplementary angles
3. $∠cpn \cong ∠lpk$	3. vertical angles theorem
4. $m∠cpn = m∠lpk$	4. definition of congruence
5. $square$	5. substitution property of equality
6. $square$	6. definition of supplementary angles
7. $overleftrightarrow{ck} \parallel overleftrightarrow{fh}$	7. $square$

Explanation:

Step 1: Analyze Step 2

The reason for step 2 is the definition of supplementary angles, which states that if two angles are supplementary, their measures add up to \(180^\circ\). So for \(\angle CPN\) and \(\angle PLH\) being supplementary, we have \(m\angle CPN + m\angle PLH = 180^\circ\).

Step 2: Analyze Step 5

We know from step 4 that \(m\angle CPN = m\angle LPK\) and from step 2 that \(m\angle CPN + m\angle PLH = 180^\circ\). Using the substitution property of equality (replacing \(m\angle CPN\) with \(m\angle LPK\)), we get \(m\angle LPK + m\angle PLH = 180^\circ\).

Step 3: Analyze Step 6

The reason for step 6 is the definition of supplementary angles, which means that if the sum of two angles' measures is \(180^\circ\), they are supplementary. So from step 5, \(\angle LPK\) and \(\angle PLH\) are supplementary (since \(m\angle LPK + m\angle PLH = 180^\circ\)).

Step 4: Analyze Step 7

To prove \(CK \parallel FH\), we can use the theorem that if two interior angles on the same side of a transversal are supplementary, then the lines are parallel. Here, \(\angle LPK\) and \(\angle PLH\) are same - side interior angles, and since they are supplementary, \(CK \parallel FH\) by the Converse of the Same - Side Interior Angles Theorem.

Answer:

1. \(m\angle CPN + m\angle PLH = 180^\circ\)
2. \(m\angle LPK + m\angle PLH = 180^\circ\)
3. \(\angle LPK\) and \(\angle PLH\) are supplementary
4. Converse of the Same - Side Interior Angles Theorem