Question

1. ∠gfh ≅ ∠ghf 1. given
2. 2. definition of congruent angles
3. 3. given (diagram)
4. 4. definition of linear pair
5. m∠efg + m∠gfh = 180° 5. definition of supplementary angles
6. 6. substitution property of equality
7. ∠efg and ∠ghf are supplementary. 7.

Explanation:

Step 2: Apply congruent angles definition

If \( \angle GFH \cong \angle GHF \), then \( m\angle GFH = m\angle GHF \) (by definition of congruent angles, congruent angles have equal measures).

Step 3: Identify linear pair (from diagram)

From the diagram (given), \( \angle EFG \) and \( \angle GFH \) form a linear pair.

Step 4: Define linear pair

A linear pair of angles are adjacent and supplementary, so \( \angle EFG \) and \( \angle GFH \) are a linear pair (meaning they are adjacent and their non - common sides form a straight line).

Step 6: Substitute angle measures

We know \( m\angle EFG + m\angle GFH = 180^\circ \) (step 5) and \( m\angle GFH = m\angle GHF \) (step 2). By substitution, \( m\angle EFG + m\angle GHF = 180^\circ \).

Step 7: Define supplementary angles

If the sum of the measures of two angles is \( 180^\circ \), then the angles are supplementary. Since \( m\angle EFG + m\angle GHF = 180^\circ \), \( \angle EFG \) and \( \angle GHF \) are supplementary (by definition of supplementary angles).

Filling the Blanks:

1. \( \boldsymbol{m\angle GFH = m\angle GHF} \)
2. \( \boldsymbol{\angle EFG} \) and \( \boldsymbol{\angle GFH} \) form a linear pair (or other diagram - based fact, e.g., \( \angle EFG \) and \( \angle GFH \) are adjacent and form a straight line)
3. \( \boldsymbol{\angle EFG} \) and \( \boldsymbol{\angle GFH} \) are a linear pair (adjacent, non - common sides form a line)
4. \( \boldsymbol{m\angle EFG + m\angle GHF = 180^\circ} \)
5. \( \boldsymbol{\text{Definition of supplementary angles}} \)

Answer:

Step 2: Apply congruent angles definition

If \( \angle GFH \cong \angle GHF \), then \( m\angle GFH = m\angle GHF \) (by definition of congruent angles, congruent angles have equal measures).

Step 3: Identify linear pair (from diagram)

From the diagram (given), \( \angle EFG \) and \( \angle GFH \) form a linear pair.

Step 4: Define linear pair

A linear pair of angles are adjacent and supplementary, so \( \angle EFG \) and \( \angle GFH \) are a linear pair (meaning they are adjacent and their non - common sides form a straight line).

Step 6: Substitute angle measures

We know \( m\angle EFG + m\angle GFH = 180^\circ \) (step 5) and \( m\angle GFH = m\angle GHF \) (step 2). By substitution, \( m\angle EFG + m\angle GHF = 180^\circ \).

Step 7: Define supplementary angles

If the sum of the measures of two angles is \( 180^\circ \), then the angles are supplementary. Since \( m\angle EFG + m\angle GHF = 180^\circ \), \( \angle EFG \) and \( \angle GHF \) are supplementary (by definition of supplementary angles).

Filling the Blanks:

1. \( \boldsymbol{m\angle GFH = m\angle GHF} \)
2. \( \boldsymbol{\angle EFG} \) and \( \boldsymbol{\angle GFH} \) form a linear pair (or other diagram - based fact, e.g., \( \angle EFG \) and \( \angle GFH \) are adjacent and form a straight line)
3. \( \boldsymbol{\angle EFG} \) and \( \boldsymbol{\angle GFH} \) are a linear pair (adjacent, non - common sides form a line)
4. \( \boldsymbol{m\angle EFG + m\angle GHF = 180^\circ} \)
5. \( \boldsymbol{\text{Definition of supplementary angles}} \)