Question

complete the proof that \\( \triangle g ij \cong \triangle igh \\).
(there is a diagram of a quadrilateral higj with a diagonal gi, and some sides marked as congruent. also, a table with statements: 1. \\( \overline{gh} \cong \overline{ij} \\), 2. \\( \overline{hi} \cong \overline{gj} \\), 3. \\( \overline{gi} \cong \overline{gi} \\), 4. \\( \triangle g ij \cong \triangle igh \\), and empty reason columns.)

Explanation:

Step1: Analyze Statement 1 ($\overline{GH} \cong \overline{IJ}$)

From the diagram, we can see that sides \( \overline{GH} \) and \( \overline{IJ} \) have the same number of tick marks, indicating they are congruent by the definition of congruent segments (segments with the same length, marked by tick marks in a diagram). So the reason for \( \overline{GH} \cong \overline{IJ} \) is "Given (or Definition of Congruent Segments from Diagram Markings)".

Step2: Analyze Statement 2 ($\overline{HI} \cong \overline{GJ}$)

Looking at the diagram, sides \( \overline{HI} \) and \( \overline{GJ} \) also have the same tick marks, so they are congruent by the same reasoning as Step 1: the diagram shows they have equal length (tick marks), so the reason is "Given (or Definition of Congruent Segments from Diagram Markings)".

Step3: Analyze Statement 3 ($\overline{GI} \cong \overline{GI}$)

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

Step4: Analyze Statement 4 ($\triangle GIJ \cong \triangle IGH$)

We have three pairs of congruent sides: \( \overline{GH} \cong \overline{IJ} \), \( \overline{HI} \cong \overline{GJ} \), and \( \overline{GI} \cong \overline{GI} \). By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So the reason for \( \triangle GIJ \cong \triangle IGH \) is "SSS Congruence Postulate".

Answer:

1. Reason for \( \overline{GH} \cong \overline{IJ} \): Congruent Segments (Diagram Markings)
2. Reason for \( \overline{HI} \cong \overline{GJ} \): Congruent Segments (Diagram Markings)
3. Reason for \( \overline{GI} \cong \overline{GI} \): Reflexive Property
4. Reason for \( \triangle GIJ \cong \triangle IGH \): SSS Congruence Postulate