Question

complete the proof that \\( \overline { f k } \cong \overline { h i } \\).

statement	reason
2 \\( \angle h j i \cong \angle f g k \\)	given
3 \\( \angle f k g \cong \angle i h j \\)	given
4 \\( g k = j k + g j \\)	additive property of length
5 \\( h j = g h + g j \\)
6 \\( g k = g h + g j \\)	substitution
7 \\( h j = g k \\)	transitive property of equality
8 \\( \triangle f g k \cong \triangle i j h \\)	asa
9 \\( \overline { f k } \cong \overline { h i } \\)	cpctc

Explanation:

Step1: Understand the Additive Property of Length

The Additive Property of Length states that if we have two line - segments \(AB\) and \(BC\) that form a longer line - segment \(AC\), then \(AC=AB + BC\). In the context of the problem, for line - segment \(HJ\), which is composed of \(GH\) and \(GJ\), we can write \(HJ=GH + GJ\) based on the Additive Property of Length.

Step2: Analyze the ASA (Angle - Side - Angle) Congruence Criterion

The ASA congruence criterion for triangles states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

• We know that \(\angle FKG\cong\angle IHJ\) (given), \(GK = HJ\) (from step 7, since \(GK=JK + GJ\), \(HJ = GH+GJ\) and \(GH\cong JK\)), and \(\angle HJI\cong\angle FGK\) (given). So, \(\triangle FGK\cong\triangle IJH\) by ASA.

Step3: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

CPCTC is a theorem that states if two triangles are congruent, then their corresponding parts (sides and angles) are congruent.
Since \(\triangle FGK\cong\triangle IJH\), and \(\overline{FK}\) and \(\overline{HI}\) are corresponding sides of the congruent triangles \(\triangle FGK\) and \(\triangle IJH\) respectively, we can conclude that \(\overline{FK}\cong\overline{HI}\) by CPCTC.

Answer:

The reason for \(HJ = GH+GJ\) is the Additive Property of Length. The proof is completed as shown above, and \(\overline{FK}\cong\overline{HI}\) is proved.