Question

step
1
statement
\\( \overline { g h } \cong \overline { e h } \\)
reason
given
2
\\( \angle c d h \cong \angle c f h \\)
in a triangle, angles opposite of congruent sides are congruent
3
\\( \angle a d h \\) and \\( \angle h d c \\) are supplementary
if two angles form a linear pair, then they are supplementary
4
type of statement
if two angles are supplements of the same angle (or congruent angles),
then they are congruent
5
\\( \angle a d h \cong \angle b f h \\)
congruent segments added to congruent segments form congruent
note: 10

Explanation:

Step1: Determine the type of statement

Since we are dealing with congruent angles (\(\angle ADH\cong\angle BFH\)) and the reason is about congruent segments added to congruent segments forming congruent segments (which is related to triangle - congruence or angle - congruence based on segment congruence), we need to check the properties.

Step2: Analyze the relationship

We know that \(\overline{GH}\cong\overline{EH}\), \(\overline{DH}\cong\overline{FH}\), \(\angle CDH\cong\angle CFH\). If we consider triangles \(GDH\) and \(EFH\) (by \(SAS\) - Side - Angle - Side, since \(\overline{GH}\cong\overline{EH}\), \(\angle GHD=\angle EHF\) (vertically - opposite angles), \(\overline{DH}\cong\overline{FH}\)), then \(\triangle GDH\cong\triangle EFH\). Then \(\angle ADH\) and \(\angle BFH\) are angles that can be shown congruent by the property that if two angles are supplements of congruent angles, they are congruent.

Answer:

1. Type of Statement: Congruent - angle statement (supplementary - angle property). Reason: If two angles are supplements of congruent angles, then they are congruent.