Question

$\overleftrightarrow{fg} \perp \overleftrightarrow{gh}$ and $\overleftrightarrow{jk} \perp \overleftrightarrow{ij}$. complete the proof that $\angle fgh \cong \angle ijk$.

	statement	reason
1	$\overleftrightarrow{fg} \perp \overleftrightarrow{gh}$	given
2	$\overleftrightarrow{jk} \perp \overleftrightarrow{ij}$	given
3	$m\angle fgh = 90^\circ$
4	$m\angle ijk = 90^\circ$
5	$m\angle fgh = m\angle ijk$
6	$\angle fgh \cong \angle ijk$	definition of congruence

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Explanation:

Step1: Reason for Statement 3

If two lines are perpendicular, the angle between them is \(90^\circ\). Since \(\overleftrightarrow{FG} \perp \overleftrightarrow{GH}\), by the definition of perpendicular lines, \(m\angle FGH = 90^\circ\).

Step2: Reason for Statement 4

Similarly, for \(\overleftrightarrow{JK} \perp \overleftrightarrow{IJ}\), using the definition of perpendicular lines, we get \(m\angle IJK = 90^\circ\).

Step3: Reason for Statement 5

Since \(m\angle FGH = 90^\circ\) and \(m\angle IJK = 90^\circ\), by the transitive property of equality (or simply because both equal \(90^\circ\)), \(m\angle FGH = m\angle IJK\).

Answer:

• For Statement 3: Definition of perpendicular lines
• For Statement 4: Definition of perpendicular lines
• For Statement 5: Transitive property of equality (or substitution, since both are \(90^\circ\))