Question

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\overline {gh}\cong \overline {jk}
\overline {ij}\cong \overline {fk}
\overline {fg}\cong \overline {hi}
gk = jk + gj
hj = gh + gj
gk = gh + gj
hj = gk
\triangle fgk\cong \triangle ihj
given
given
given
additive property of length
additive property of length
all right angles are congruent
definition of angle bisector
definition of equilateral triangle
definition of midpoint
vertical angle theorem

Explanation:

Step1: Analyze the given information

We have \( \overline{GH}\cong\overline{JK}\), \( \overline{IJ}\cong\overline{FK}\), \( \overline{FG}\cong\overline{HI}\), \( GK = JK+GJ\) (by the Additive Property of Length which states that if we have three collinear points \(J\), \(K\), \(G\) then \(GK=GJ + JK\)), \(HJ=GH + GJ\) (by the Additive Property of Length which states that if we have three collinear points \(G\), \(H\), \(J\) then \(HJ=GH + GJ\)).

Step2: Substitute \(JK\) with \(GH\)

Since \( \overline{GH}\cong\overline{JK}\), then \(JK = GH\). Substituting \(JK\) in the equation \(GK = JK+GJ\) gives \(GK=GH + GJ\).

Step3: Use the transitive property

We know that \(HJ=GH + GJ\) and \(GK=GH + GJ\). By the transitive property of equality (if \(a=b\) and \(c = b\), then \(a=c\)), we can conclude that \(HJ = GK\).

Answer:

The reason for step 6 (\(GK=GH + GJ\)) and step 5 (\(HJ=GH + GJ\)) is the Additive Property of Length. The reason for step 7 (\(HJ = GK\)) is the transitive property of equality.