Question

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

statement	reason
2. m∠cpn + m∠hlp = 180°	2. definition of supplementary angles
3. ∠cpn ≅ ∠lpk	3. vertical angles theorem
4. m∠cpn = m∠lpk	4. definition of congruence
5.	5. substitution property of equality
6.	6. definition of supplementary angles
7. \overleftrightarrow{ck} \parallel \overleftrightarrow{fh}	7.

Explanation:

Step 1: Analyze Step 5 (Substitution)

We know from step 4 that \( m\angle CPN = m\angle LPK \), and from step 2 that \( m\angle CPN + m\angle HLP = 180^\circ \). By substitution (replacing \( m\angle CPN \) with \( m\angle LPK \) in the equation from step 2), we get:
\( m\angle LPK + m\angle HLP = 180^\circ \)

Step 2: Analyze Step 6 (Definition of Supplementary)

The equation \( m\angle LPK + m\angle HLP = 180^\circ \) means \( \angle LPK \) and \( \angle HLP \) are supplementary (by the definition of supplementary angles: two angles whose measures add up to \( 180^\circ \) are supplementary).

Step 3: Analyze Step 7 (Prove Parallel)

\( \angle LPK \) and \( \angle HLP \) are same - side interior angles (they lie between the two lines \( CK \) and \( FH \) and on the same side of the transversal \( PL \)). If same - side interior angles are supplementary, then the two lines are parallel (Converse of the Same - Side Interior Angles Theorem). So, since \( \angle LPK \) and \( \angle HLP \) are supplementary, \( CK\parallel FH \).

Filling in the Table:

• Step 5 Statement: \( m\angle LPK + m\angle HLP = 180^\circ \)
• Step 6 Statement: \( \angle LPK \) and \( \angle HLP \) are supplementary
• Step 7 Reason: Converse of the Same - Side Interior Angles Theorem (If two lines are cut by a transversal and the same - side interior angles are supplementary, then the lines are parallel)

Answer:

Step 5: \( \boldsymbol{m\angle LPK + m\angle HLP = 180^\circ} \)

Step 6: \( \boldsymbol{\angle LPK} \) and \( \boldsymbol{\angle HLP} \) are supplementary

Step 7: Converse of the Same - Side Interior Angles Theorem