Question

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

Explanation:

Step1: Recall triangle congruence criteria

ASA (Angle - Side - Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
AAS (Angle - Angle - Side): If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent.
HL (Hypotenuse - Leg): For right - angled triangles, if the hypotenuse and a leg are congruent, the triangles are congruent.
SAS (Side - Angle - Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.

Step2: Identify given congruent parts

We know \(AF\cong ED\) (given), \(\angle AFC\cong\angle DEB\) (all right angles are congruent), and \(AC\cong BD\) (from previous steps in the proof).
The triangles in question are \(\triangle AFC\) and \(\triangle DEB\). We have two sides (\(AF\cong ED\), \(AC\cong BD\)) and the included right angles (\(\angle AFC\cong\angle DEB\)) not. But wait, if we consider the right - angled triangles \(\triangle AFC\) and \(\triangle DEB\), we have \(AF = ED\) (from \(AF\cong ED\)), \(AC=BD\) (from \(AC\cong BD\)) and \(\angle AFC=\angle DEB = 90^{\circ}\).
For right - angled triangles, the HL (Hypotenuse - Leg) criterion: In right - triangles \(\triangle AFC\) and \(\triangle DEB\), \(AC\) and \(BD\) are hypotenuses (\(AC = BD\)) and \(AF\) and \(ED\) are legs (\(AF=ED\)).

Answer:

HL