Question

complete the proof that \\(\overleftrightarrow{gi} \perp \overleftrightarrow{fh}\\).

statements

1. \\(\angle fjg \cong \angle gjh\\)
2. \\(m\angle fjg + m\angle gjh = 180^\circ\\)
3. \\(m\angle fjg + m\angle fjg = 180^\circ\\)
4. \\(m\angle fjg = 90^\circ\\)
5. \\(\overleftrightarrow{gi} \perp \overleftrightarrow{fh}\\)

select an answer
given
definitions of supplementary angles
definition of congruence
algebra
transitive property of equality
transitive property of congruence
definition of complementary angles
substitution
angles forming a linear pair sum to 180
definition of perpendicular lines
reflexive property of equality
reflexive property of congruence
definition of angle bisector

Explanation:

Step1: Match statement 1 to reason

$\angle FJG \cong \angle GJH$ is a given premise for the proof.

Step2: Match statement 2 to reason

$\angle FJG$ and $\angle GJH$ form a linear pair, so their measures sum to $180^\circ$.
$m\angle FJG + m\angle GJH = 180^\circ$

Step3: Match statement 3 to reason

Substitute $\angle GJH$ with $\angle FJG$ (from congruence).
$m\angle FJG + m\angle FJG = 180^\circ$

Step4: Match statement 4 to reason

Simplify the equation algebraically.
$2m\angle FJG = 180^\circ \implies m\angle FJG = 90^\circ$

Step5: Match statement 5 to reason

A right angle means the lines are perpendicular.
$\overline{GI} \perp \overline{FH}$

Answer:

1. Given
2. Angles forming a linear pair sum to 180
3. Substitution
4. Algebra
5. Definition of perpendicular lines