Question

justifying a statement
based on the diagram, which reason justifies this
statement?
if \\( \angle c e f \\) is complementary to \\( \angle d c f \\), then
\\( \angle d c f \cong \angle f e g \\).
complements of the same angle are congruent.
complements of congruent angles are congruent.
supplements of the same angle are congruent.
supplements of congruent angles are congruent.

Explanation:

• First, note that \(\angle CEF+\angle FEG = 90^{\circ}\) (since \(\angle CEG = 90^{\circ}\) as it is a right - angle). So \(\angle FEG\) is the complement of \(\angle CEF\).
• Given that \(\angle CEF\) is complementary to \(\angle DCF\), which means \(\angle CEF+\angle DCF=90^{\circ}\).
• Let \(x = \angle CEF\). Then \(\angle FEG=90 - x\) and \(\angle DCF = 90 - x\) (because if \(x + y=90\) and \(x+z = 90\), then \(y = z\)). Here, \(\angle CEF\) is the same angle for both \(\angle DCF\) and \(\angle FEG\) (they are both complements of \(\angle CEF\)).

Answer:

Complements of the same angle are congruent.