Question

Which two statements, if true, guarantee that FG ↔ AB is perpendicular to AB ↔ CD?

(A) AB ↔ CD is a straight line, ∠ABC = ∠CDF
(B) AB ↔ CD is a straight line, ∠ABC = ∠BCD
(C) AB ↔ CD is a straight line, ∠FBC = ∠BCD
(D) AB ↔ CD is a straight line, ∠ABC = ∠FBC

Answer 1

Option (B) guarantees that line FG is perpendicular to AB because if ∠ABC = ∠BCD and AB↔CD is a straight line, each angle would be 90°, indicating perpendicularity.

Step-by-step explanation:

To determine which statements guarantee that line FG is perpendicular to the line segment AB, we need to consider the relationships between angles and lines. A right angle is formed when two lines are perpendicular, creating a 90° angle. Therefore, the statements that involve the angles adding up to 90° would be the ones that guarantee perpendicularity.

Looking at the options given: Option (B) states that AB↔CD is a straight line, and ∠ABC = ∠BCD. If these angles are equal, and we know that a straight line creates a 180° angle, each of these angles must be 90°. Therefore, option (B) would guarantee that FG↔AB is perpendicular to AB↔CD because it suggests FG↔AB intersecting line AB↔CD would split the straight line into two equal right angles.

Option (D) is mentioning that ∠ABC is equal to ∠FBC. However, there is no guarantee about the measure of these angles, so it doesn't necessarily imply perpendicularity.

As for options (A) and (C), they do not provide enough information to conclude that the angles involved are right angles without additional context. Hence, the clear answer here is option (B).