Question

∠gjh ≅ ∠eik, ∠k ≅ ∠h, and \overline{jk} ≅ \overline{hi}. complete the proof that \triangle ghj ≅ \triangle eki.

statement	reason
2	∠k ≅ ∠h	given
3	\overline{jk} ≅ \overline{hi}	given
4	ik = jk + ij	additive property of length
5	hj = hi + ij	additive property of length
6	ik = hi + ij	substitution
7	hj = ik
8	\triangle ghj ≅ \triangle eki	asa

Explanation:

Step1: Analyze the equations for \(IK\) and \(HJ\)

From step \(4\): \(IK = JK+IJ\), from step \(3\): \(JK\cong HI\) (so \(JK = HI\)), then \(IK=HI + IJ\) (substitution). From step \(5\): \(HJ=HI + IJ\).

Step2: Use substitution property

Since \(IK = HI+IJ\) and \(HJ=HI + IJ\), by the substitution property (if \(a=b\) and \(c = b\), then \(a=c\)), we can say \(HJ = IK\).

Answer:

Substitution (because \(IK = HI+IJ\) and \(HJ=HI + IJ\), so \(HJ = IK\) by substituting \(IK\) with \(HI + IJ\) in the equation for \(HJ\))