Question

A 4-sided figure has 4 corners labeled E, G, D, C starting from the top-left corner and going clockwise. A line cuts through the shape from point C to point G and contains point F. Another line cuts through the shape from E to point F. Angle C E G is a right angle.
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.

Answer 1

To justify the statement that if ∠CEF is complementary to ∠DCF, then ∠DCF ≅ ∠FEG, the correct reasoning to use is **"Complements of the same angle are congruent."**

Here's the breakdown of the reasoning:

1. Since ∠CEF is complementary to ∠DCF, this means that together, they add up to 90 degrees.

2. If ∠CEF is complementary to ∠DCF, then any angle that is complementary to ∠CEF (let's consider ∠FEG) must also coincide with the relationship involving ∠DCF.

3. Therefore, if two angles are each complementary to the same angle (in this case, ∠CEF), they must be congruent to each other.

Thus, the appropriate justification for the statement given is "Complements of the same angle are congruent."