Question

name all of the angle(s) in the diagram that are adjacent and congruent to \\(\angle bdc\\)

choose the correct answer(s) below. select all that apply.

a. \\(\angle gfd\\)
b. \\(\angle bdg\\)
c. \\(\angle gdf\\)
d. \\(\angle hdf\\)
e. \\(\angle hde\\)
f. there are no angles adjacent and congruent to \\(\angle bdc\\)

Explanation:

Identify adjacent angles to \(\angle BDC\)

The angles sharing vertex \(D\) and ray \(DB\) or ray \(DC\) without overlapping interiors are:

• \(\angle BDF\) (sharing ray \(DB\))
• \(\angle CDE\) (sharing ray \(DC\))

Determine congruency based on geometric markings

• \(\angle BDC\) has a single arc marking.
• \(\angle CDE\) has a right-angle square marking, so \(m\angle CDE = 90^\circ\).
• \(\angle BDF\) has a single arc marking, indicating \(\angle BDF \cong \angle BDC\).
• Note that the line containing \(C, D, F\) is a straight line, so \(D\) is the intersection. The ray opposite to \(DC\) is \(DF\). Thus, \(\angle BDF\) is adjacent to \(\angle BDC\) along the line \(BH\).

Match with given options

• The adjacent and congruent angle is \(\angle BDF\) (or \(\angle GDB\), since \(G, D, E\) lie on the same line, making ray \(DG\) the same as ray \(DF\) in terms of the angle with \(DB\)).
• Looking at the options:
• A. \(\angle GFD\)
• B. \(\angle BDG\)
• C. \(\angle GDF\)
• D. \(\angle HDF\)
• E. \(\angle HDE\)
• F. There are no angles adjacent and congruent to \(\angle BDC\).
• Since ray \(DF\) passes through \(G\) (the line is \(EG\)), \(\angle BDF\) is the same angle as \(\angle BDG\).
• Therefore, \(\angle BDG\) is adjacent to \(\angle BDC\) and congruent to it.

Answer:

• A. \(\angle GFD\)
• B. \(\angle BDG\) (Correct answer)
• C. \(\angle GDF\)
• D. \(\angle HDF\)
• E. \(\angle HDE\)
• F. There are no angles adjacent and congruent to \(\angle BDC\).