Question

fill in the missing statement and reason of the proof below.
given: \\( \overline { g h } \cong \overline { e h } \\), \\( \overline { d h } \cong \overline { f h } \\) and \\( \angle c d h \cong \angle c f h \\).
prove: \\( \angle a \cong \angle b \\).

step	statement	reason
2	\\( \overline { d h } \cong \overline { f h } \\)	given
3	\\( \angle c d h \cong \angle c f h \\)	given
4	\\( \angle h g e \cong \angle h e b \\)	in a triangle angles opposite of congruent sides are congruent
5	\\( \angle a d h \\) and \\( \angle h d c \\) are supplementary	if two angles form a linear pair, then they are supplementary
6	\\( \angle a d h \cong \angle b f h \\)	if two angles are supplements of the same angle (or congruent angles), then they are congruent
7	\\( \overline { g f } \cong \overline { e d } \\)	congruent segments added to congruent segments form congruent segments

Explanation:

Step1: Identify congruent triangles

Since \(GH\cong EH\), \(DH\cong FH\), and \(\angle GHD=\angle EHF\) (vertical angles), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle GHD\cong\triangle EHF\).

Step2: Use triangle congruence properties

If \(\triangle GHD\cong\triangle EHF\), then \(\angle ADH\cong\angle BFH\) (corresponding parts of congruent triangles).

Step3: Consider supplementary angles

\(\angle ADH\) and \(\angle HDC\) are supplementary, \(\angle BFH\) and \(\angle HFC\) are supplementary. Since \(\angle ADH\cong\angle BFH\), then \(\angle HDC\cong\angle HFC\) (if two angles are congruent, their supplements are congruent).

Step4: Use angle - angle - side (AAS) or other congruence

In \(\triangle ADH\) and \(\triangle BFH\), we have \(DH = FH\), \(\angle ADH=\angle BFH\), and \(\angle AHD=\angle BHF\) (vertical angles). By AAS, \(\triangle ADH\cong\triangle BFH\).

Step5: Conclude angle congruence

Since \(\triangle ADH\cong\triangle BFH\), then \(\angle A\cong\angle B\) (corresponding parts of congruent triangles).

Answer:

For step 4: \(\triangle GHD\cong\triangle EHF\) (by SAS, since \(GH = EH\), \(DH=FH\), \(\angle GHD=\angle EHF\) (vertical angles)); for step 4 reason: Side - Angle - Side (SAS) congruence criterion.