Question

melissa wants to prove that if a transversal crosses parallel lines, the same - side interior angles are supplementary. she uses this diagram, in which ( overleftrightarrow{ab}paralleloverleftrightarrow{cd} ).
complete melissas proof that same - side interior angles ( angle afg ) and ( angle cgf ) are supplementary.

statement	reason

|2 ( angle afgcongangle cgh )|

3 ( mangle afg = mangle cgh )	definition of congruence
4 ( \text{ }+mangle cgf = 180^{circ} )	angles forming a linear pair sum to ( 180^{circ} ).
5 ( \text{ }+mangle cgf = 180^{circ} )	substitution property of equality
6 ( mangle afg ) and ( mangle cgf ) are supplementary.	definition of supplementary angles

Explanation:

Step1: Identify the reason for congruent angles

When two parallel lines are cut by a transversal, corresponding angles are congruent. Here, \(\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}\) and \(EH\) is the transversal. \(\angle AFG\) and \(\angle CGH\) are corresponding angles. So the reason for \(\angle AFG\cong\angle CGH\) is "Corresponding Angles Postulate".

Step2: Determine the angle for the linear - pair equation

Since \(\angle CGH\) and \(\angle CGF\) form a linear pair (they are adjacent angles whose non - common sides are opposite rays), the angle in the equation \(m\angle\square + m\angle CGF=180^{\circ}\) (from the reason "Angles forming a linear pair sum to \(180^{\circ}\)") is \(m\angle CGH\).

Step3: Substitute using the congruent angles

Because \(m\angle AFG = m\angle CGH\) (from step 3, definition of congruence), when we substitute \(m\angle CGH\) with \(m\angle AFG\) in the equation \(m\angle CGH+m\angle CGF = 180^{\circ}\), the equation in step 5 becomes \(m\angle AFG+m\angle CGF = 180^{\circ}\).

Answer:

1. Corresponding Angles Postulate; 4. \(m\angle CGH\); 5. \(m\angle AFG\)