Question

complete the proof that ∠eif≅∠gih.
statement reason
1 m∠eif + m∠fig = 180° angles forming a linear pair sum to 180°
2 m∠fig + m∠gih = 180°
3 m∠eif + m∠fig = m∠fig + m∠gih properties of addition, subtraction, multiplication, and division
4 m∠eif = m∠gih definition of congruence
5 ∠eif≅∠gih

Explanation:

Step1: Identify linear - pair angles

$m\angle EIF + m\angle FIG=180^{\circ}$ because $\angle EIF$ and $\angle FIG$ form a linear pair. Also, $m\angle FIG + m\angle GIH = 180^{\circ}$ since $\angle FIG$ and $\angle GIH$ form a linear pair.

Step2: Set equal expressions

Since $m\angle EIF + m\angle FIG=180^{\circ}$ and $m\angle FIG + m\angle GIH = 180^{\circ}$, by the transitive property of equality (if $a = c$ and $b = c$, then $a = b$), we can say $m\angle EIF + m\angle FIG=m\angle FIG + m\angle GIH$.

Step3: Subtract $m\angle FIG$ from both sides

Using the subtraction property of equality (if $a + b=c + b$, then $a = c$), subtracting $m\angle FIG$ from both sides of $m\angle EIF + m\angle FIG=m\angle FIG + m\angle GIH$ gives $m\angle EIF=m\angle GIH$.

Step4: Use congruence definition

By the definition of congruent angles (if the measures of two angles are equal, then the angles are congruent), since $m\angle EIF=m\angle GIH$, we have $\angle EIF\cong\angle GIH$.

Answer:

The proof is completed as shown above, showing that $\angle EIF\cong\angle GIH$ by using properties of linear - pair angles, equality, and the definition of congruent angles.