Question

analyze the figure and the two - column proof.
which statement/reason pair can be used when proving ( overline{fc} cong overline{eb} )?

Explanation:

Step1: Identify congruent triangles

We know that \(\overline{AF}\cong\overline{ED}\), \(\angle AFC\cong\angle DEB\) (right - angles), and \(\overline{AC}\cong\overline{BD}\) (from previous steps in the proof where \(AC = BD\) and the definition of congruent segments).
By the Hypotenuse - Leg (HL) congruence criterion for right - triangles (if the hypotenuse and one leg of a right - triangle are congruent to the hypotenuse and one leg of another right - triangle, then the two right - triangles are congruent).
In right - triangles \(\triangle AFC\) and \(\triangle DEB\), \(\overline{AF}\) and \(\overline{ED}\) are legs, \(\overline{AC}\) and \(\overline{BD}\) are hypotenuses.
So, \(\triangle DEB\cong\triangle AFC\) (by HL criterion).

Step2: Use congruent triangle properties

If \(\triangle DEB\cong\triangle AFC\), then their corresponding parts are congruent. Since \(\overline{FC}\) and \(\overline{EB}\) are corresponding parts of \(\triangle AFC\) and \(\triangle DEB\) respectively.
The reason is that corresponding parts of congruent triangles are congruent (CPCTC).

Answer:

1. \(\overline{FC}\cong\overline{EB}\); 9. Corresponding parts of congruent triangles are congruent (CPCTC)