In a rectangle where opposite sides are congruent and diagonals intersect at a point G, segments connecting the point G to opposite vertices (EB and EC) are also congruent.
1. Properties of Rectangles:
Opposite sides are congruent (AB ≅ CD)
Diagonals bisect each other (AG ≅ BG and CG ≅ DG)
2. Triangle Similarity:
Triangles ABG and CDG share two pairs of congruent sides (AB ≅ CD and AG ≅ CG) and have a common included angle (∠AGB ≅ ∠CDG, right angles due to rectangle definition).
By AA Similarity, triangles ABG and CDG are similar.
3. Corresponding Side Congruence:
Since triangles ABG and CDG are similar, corresponding sides have equal ratios.
Specifically, EB/EC = AG/CG = 1 (as AG ≅ CG due to diagonal bisection).
Therefore, EB ≅ EC (multiplying both sides by EC).
Therefore, in rectangle ABCD with diagonals intersecting at G and FA ≅ GD, EB is congruent to EC.