Question

∠fig≅∠eif. complete the proof that (overleftrightarrow{eg}perpoverleftrightarrow{fh}).
statement\treason
1 ∠fig≅∠eif
2 (mangle eif + mangle fic) paste autofill ing a linear pair sum to 180°
3 (mangle eif + mangle eif = 180^{circ})
4 (mangle eif = 90^{circ}) properties of addition, subtraction, multiplication, and division reflexive property of congruence reflexive property of equality substitution transitive property of congruence transitive property of equality
5 (overleftrightarrow{eg}perpoverleftrightarrow{fh})

Explanation:

Step1: Recall linear - pair property

$\angle EIF$ and $\angle FIG$ are a linear pair, so $m\angle EIF + m\angle FIG=180^{\circ}$

Step2: Use angle - congruence substitution

Since $\angle FIG\cong\angle EIF$, then $m\angle EIF = m\angle FIG$. Substituting $m\angle FIG$ with $m\angle EIF$ in $m\angle EIF + m\angle FIG = 180^{\circ}$, we get $m\angle EIF+m\angle EIF = 180^{\circ}$

Step3: Solve for $m\angle EIF$

Combining like - terms, $2m\angle EIF=180^{\circ}$. Dividing both sides by 2 gives $m\angle EIF = 90^{\circ}$

Step4: Define perpendicular lines

If the measure of the angle formed by two lines is $90^{\circ}$, the two lines are perpendicular. Since $\angle EIF$ is the angle formed by $\overleftrightarrow{EG}$ and $\overleftrightarrow{FH}$ and $m\angle EIF = 90^{\circ}$, then $\overleftrightarrow{EG}\perp\overleftrightarrow{FH}$

Answer:

1. Given
2. Definition of a linear pair
3. Substitution
4. Properties of addition, subtraction, multiplication, and division
5. Definition of perpendicular lines