Question

based on the diagram, which reason justifies this statement? if ∠cef is complementary to ∠dcf, then ∠dcf ≅ ∠feg 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:

• We know that \(\angle CEF+\angle DCF = 90^{\circ}\) (given that \(\angle CEF\) is complementary to \(\angle DCF\))
• Also, in right - triangle \(CEG\) (\(\angle CEG = 90^{\circ}\)), \(\angle CEF+\angle FEG=90^{\circ}\)
• Let \(\angle A=\angle CEF\). Then \(\angle DCF = 90^{\circ}-\angle A\) and \(\angle FEG=90^{\circ}-\angle A\)
• By the definition of complementary angles (\(\alpha+\beta = 90^{\circ}\), \(\beta\) is the complement of \(\alpha\)), \(\angle DCF\) and \(\angle FEG\) are both complements of \(\angle CEF\)
• According to the theorem “Complements of the same angle are congruent” (if \(\angle x+\angle y = 90^{\circ}\) and \(\angle x+\angle z=90^{\circ}\), then \(\angle y\cong\angle z\)), we can conclude that \(\angle DCF\cong\angle FEG\)

Answer:

Complements of the same angle are congruent