Question

i. ( mangle b = mangle f ), ii. ( bc = fg ), iii. ( mangle a = mangle e ), iv. ( fg = 3 ), v. ( mangle b = mangle e ). options: a. i or iv only, b. i and iii only, c. i only, d. iii or v only.

Explanation:

Step1: Analyze Triangle Congruence

From the diagram, triangles \( \triangle ABC \) and \( \triangle EFG \) seem to have congruent sides (e.g., \( AB = EF = 2 \), \( BC = FG = 3 \)) and right angles ( \( \angle B \) and \( \angle F \) or \( \angle E \)? Wait, \( \angle B \) is a right angle? Wait, let's check the angles. If \( \triangle ABC \cong \triangle EFG \) (by SAS, since \( AB = EF \), \( BC = FG \), and included angles \( \angle B \) and \( \angle F \) or \( \angle E \)? Wait, maybe \( \angle B = \angle F \) (right angles) or \( \angle B = \angle E \)? Wait, the options:

i. \( m\angle B = m\angle F \): If \( \triangle ABC \cong \triangle EFG \), then corresponding angles are equal. If \( AB = EF \), \( BC = FG \), and included angle \( \angle B \) and \( \angle F \), then i is true.

iv. \( FG = 3 \): From the diagram, \( BC = 3 \), and if \( BC = FG \) (congruent sides), then iv is true. So i or iv (option A) makes sense.

Step2: Evaluate Other Options

• Option B: i and iii. \( m\angle A = m\angle E \): Not necessarily corresponding unless congruence is by another criterion, but we have sides \( AB, BC \) and \( EF, FG \), so included angle is \( \angle B \) and \( \angle F \), so iii may not hold.
• Option C: i only: But iv (FG=3) is also true as BC=3 and congruent.
• Option D: iii or v: \( m\angle A = m\angle E \) or \( m\angle B = m\angle E \): Not correct as \( \angle B \) should correspond to \( \angle F \) if SAS with \( AB, BC \) and \( EF, FG \).

Answer:

A. i or iv only