Question

$\overline{gi} \parallel \overline{fj}$. complete the proof that $m\angle f + m\angle j + m\angle fgj = 180^\circ$ without using the triangle angle sum theorem.

(image of a geometric figure with points f, j, g, i, h and a table with statements and reasons for a proof.)

1. $\overline{gi} \parallel \overline{fj}$ - given
2. $\angle f \cong \angle hgi$ - corresponding angles theorem
3. $\angle j \cong \angle igj$ - alternate interior angles theorem
4. $m\angle hgj = m\angle hgi + m\angle igj$ - additive property of angle measure
5. $m\angle hgj = m\angle f + m\angle j$ - substitution
6. $m\angle hgj + m\angle fgj = 180^\circ$ - reason to be filled
7. $m\angle f + m\angle j + m\angle fgj = 180^\circ$ - reason to be filled

Explanation:

Step1: Analyze Statement 6

We need to find the reason for \( m\angle HGJ + m\angle FGJ = 180^\circ \). Notice that \( \angle HGJ \) and \( \angle FGJ \) form a linear pair (they are adjacent angles that form a straight line). By the definition of a linear pair, adjacent angles forming a straight line are supplementary, meaning their measures add up to \( 180^\circ \). So the reason for statement 6 is "Linear Pair Postulate" (or "Definition of Linear Pair").

Step2: Analyze Statement 7

For statement 7, we substitute \( m\angle HGJ \) with \( m\angle F + m\angle J \) (from statement 5) into the equation from statement 6. So we use the Substitution Property again.

Answer:

• For statement 6: Reason is "Linear Pair Postulate" (or "Definition of Linear Pair", as adjacent angles forming a straight line are supplementary).
• For statement 7: Reason is "Substitution" (substituting \( m\angle HGJ \) with \( m\angle F + m\angle J \) from statement 5 into \( m\angle HGJ + m\angle FGJ = 180^\circ \)).