Question

given:
\\(\overline{fh} \parallel \overline{ik}\\)
\\(\overline{gf} \cong \overline{ji}\\)
\\(\overline{fh} \cong \overline{ik}\\)

prove: \\(\delta gfh \cong \delta jik\\)

Explanation:

⚡ Using what you learned: proofs involving angles, segments, triangles, and parallelograms

Step 1: Identify the missing angle relationship

The parallel lines \(\overline{FH} \parallel \overline{IK}\) are cut by the transversal line containing points \(G\), \(F\), \(J\), and \(I\).

This transversal creates corresponding angles at the intersections:

$$ \angle GFH \cong \angle JIK $$

The reason for this statement is the Corresponding Angles Postulate.

Step 2: Determine the triangle congruence criteria

We are given:

1. \(\overline{GF} \cong \overline{JI}\) (Side)
2. \(\angle GFH \cong \angle JIK\) (Included Angle)
3. \(\overline{FH} \cong \overline{IK}\) (Side)

Using these three pieces of information, the triangles are congruent by the SAS (Side-Angle-Side) Congruence Postulate.

Step 3: Identify the final CPCTC statement

Once the triangles are proven congruent (\(\Delta GFH \cong \Delta JIK\)), any remaining corresponding parts are congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

The remaining corresponding sides or angles that can be concluded are:

• \(\overline{GH} \cong \overline{JK}\)
• \(\angle G \cong \angle J\)
• \(\angle H \cong \angle K\)

Typically, in these geometric proofs, we prove the congruence of the third side:

$$ \overline{GH} \cong \overline{JK} $$

Answer:

• First empty box (under \(\overline{FH} \parallel \overline{IK}\)):
• Statement: \(\angle GFH \cong \angle JIK\)
• Reason: Corresponding Angles Postulate

• Reason box under \(\Delta GFH \cong \Delta JIK\):
• Reason: SAS Congruence Postulate

• Bottom empty box (above CPCTC):
• Statement: \(\overline{GH} \cong \overline{JK}\) (or \(\angle G \cong \angle J\), or \(\angle H \cong \angle K\))