Question

in addition to the facts in the diagram, which other statements are necessary to prove that δabc is congruent to δefg 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. i or iv only
b. i only
c. i and iii only
d. iii or v only

Explanation:

Step1: Recall ASA Criterion

ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

From the diagram, we can see that \(AB = FE = 2\) (given by the length markings). Let's analyze each statement:

Step2: Analyze Statement i (\(m\angle B=m\angle F\))

If \(m\angle B = m\angle F\), and we know \(AB=FE\), and if we can get another pair of angles (e.g., \(\angle A=\angle E\) or \(\angle C=\angle G\)) related to the included side, but let's check statement iii (\(m\angle A = m\angle E\))

Step3: Analyze Statement iii (\(m\angle A=m\angle E\))

If we have \(AB = FE\) (side), \(m\angle B=m\angle F\) (angle) and \(m\angle A = m\angle E\) (angle), then by ASA (since \(AB\) is included between \(\angle A\) and \(\angle B\), and \(FE\) is included between \(\angle E\) and \(\angle F\)), \(\triangle ABC\cong\triangle EFG\).

Now let's check the other statements:

• Statement ii (\(BC = FG\)): This is a side - side - angle or side - angle - side related, not ASA.
• Statement iv (\(FG = 3\)): Just a length, not related to angles for ASA.
• Statement v (\(m\angle B=m\angle E\)): The angles are not the included angles with the given side \(AB = FE\).

So we need either statement i and iii together. Let's check the options:

Option C says "i and iii only", which matches our analysis.

Answer:

C. i and iii only