Question

in addition to the facts in the diagram, which other statements are necessary to prove that ?abc is congruent to ?efg by the asa criterion?

i. ( mangle b=mangle f )
ii. ( bc = fg )
iii. ( mangle a=mangle e )
iv. ( fg = 3 )
v. ( mangle b=mangle e )

a. iii or v only
b. i only
c. i or iv only
d. i and ii only

Explanation:

To prove \(\triangle ABC \cong \triangle EFG\) by ASA (Angle - Side - Angle) criterion, we need two angles and the included side to be equal. From the diagram, we know that \(AB = EF = 2\).

• For ASA, we need the angles adjacent to this side to be equal. In \(\triangle ABC\), the angles adjacent to \(AB\) are \(\angle A\) and \(\angle B\). In \(\triangle EFG\), the angles adjacent to \(EF\) are \(\angle E\) and \(\angle F\).
• If we consider statement i: \(m\angle B=m\angle F\), and we can also get \(m\angle A = m\angle E\) (statement iii) or check the other conditions. Wait, let's re - evaluate. The side \(AB = EF = 2\). For ASA, the included side is between two angles. So we need \(\angle B=\angle F\) (statement i) and \(\angle A=\angle E\) (statement iii) because \(AB\) is between \(\angle A\) and \(\angle B\), and \(EF\) is between \(\angle E\) and \(\angle F\). Let's check the options:
• Option A: iii or v only. Statement v is \(m\angle B = m\angle E\), which is not the correct angle for ASA as \(\angle B\) should correspond to \(\angle F\) (since \(AB\) and \(EF\) are equal and are the sides between the angles). So A is wrong.
• Option B: i only. If we only have \(\angle B=\angle F\), we still need \(\angle A=\angle E\) because ASA requires two angles and the included side. So B is wrong.
• Option C: i or iv only. Statement iv is \(FG = 3\), and \(BC = 3\) from the diagram. But \(FG = 3\) is related to SSS or SAS, not ASA. So C is wrong.
• Option D: i and iii only. Statement i gives \(\angle B=\angle F\) and statement iii gives \(\angle A=\angle E\), and we know \(AB = EF\). So by ASA (\(\angle A=\angle E\), \(AB = EF\), \(\angle B=\angle F\)), we can prove the triangles are congruent.

Answer:

D. i and iii only