Question

complete the proof that \\( \triangle gji \cong \triangle igh \\).
(image of a quadrilateral with vertices g, h, i, j and a diagonal gi. segments gh and ij are marked with two red ticks, segments hi and gj are marked with one red tick.)

statement	reason
2. \\( \overline{hi} \cong \overline{gj} \\)
3. \\( \overline{gi} \cong \overline{gi} \\)
4. \\( \triangle gji \cong \triangle igh \\)	sss

Explanation:

Step1: Analyze Statement 1

We need to find the reason why \(\overline{GH} \cong \overline{IJ}\). From the diagram, the sides \(GH\) and \(IJ\) have the same number of tick marks, indicating they are congruent by the definition of congruent segments (or because the figure appears to be a parallelogram, so opposite sides are congruent). So the reason is "Given (or Opposite sides of a parallelogram are congruent)".

Step2: Analyze Statement 2

For \(\overline{HI} \cong \overline{GJ}\), similarly, from the diagram, \(HI\) and \(GJ\) have the same tick marks. The reason is also "Given (or Opposite sides of a parallelogram are congruent)".

Step3: Analyze Statement 3

\(\overline{GI} \cong \overline{GI}\) is the reflexive property of congruence, which states that any segment is congruent to itself. So the reason is "Reflexive Property of Congruence".

Step4: Analyze Statement 4

We have three pairs of congruent sides (\(\overline{GH} \cong \overline{IJ}\), \(\overline{HI} \cong \overline{GJ}\), \(\overline{GI} \cong \overline{GI}\)), so by the SSS (Side - Side - Side) Congruence Postulate, \(\triangle G IJ\cong\triangle IGH\). The reason is "SSS Congruence Postulate".

Answer:

1. Reason: Given (or Opposite sides of a parallelogram are congruent)
2. Reason: Given (or Opposite sides of a parallelogram are congruent)
3. Reason: Reflexive Property of Congruence
4. Reason: SSS Congruence Postulate