Question

which of these statements would prove that line segment eb is perpendicular to line segment fd? diagram: lines intersect at b; vertical line with a (top), c (bottom); other lines: f (left), e (top-right), d (bottom-right). options: angle fba and angle abe add up to 90 degrees, angle fbe and angle ebd add up to 180 degrees, angle ebd is congruent to angle fbe, angle dbc and angle ebd add up to 90 degrees

Explanation:

To determine which statement proves \( EB \perp FD \), we analyze each option:

1. Option 1 (Angle FBA and Angle ABE add to \( 90^\circ \)):

These angles are adjacent along \( BA \), but this does not relate to \( FD \) or perpendicularity.

1. Option 2 (Angle FBE and Angle EBD add to \( 180^\circ \)):

This would mean \( FBE \) and \( EBD \) are supplementary (form a straight line), but perpendicularity requires a \( 90^\circ \) angle, not \( 180^\circ \).

1. Option 3 (Angle EBD ≅ Angle FBE):

If \( \angle EBD \cong \angle FBE \), and assuming \( \angle FBE + \angle EBD = 180^\circ \) (linear pair), then each angle would be \( 90^\circ \) (since \( \angle FBE = \angle EBD \) and their sum is \( 180^\circ \), so \( \angle FBE = \angle EBD = 90^\circ \)). This implies \( EB \perp FD \) (since the angle between them is \( 90^\circ \)).

1. Option 4 (Angle DBC and Angle EBD add to \( 90^\circ \)):

These angles involve \( BC \), not directly related to \( FD \) or the angle between \( EB \) and \( FD \).

Answer:

The correct statement is "Angle EBD is congruent to Angle FBE" (the third option, depending on the original labeling of checkboxes; typically, this corresponds to the option stating \( \angle EBD \cong \angle FBE \)).