Question

1. sas

a) additional known statement:

reason:

b) information still needed:

to prove the δ ≅ by sas

Explanation:

Step1: Analyze the given figure and the SAS (Side - Angle - Side) congruence criterion

We know that for two triangles to be congruent by SAS, two sides and the included angle of one triangle must be equal to the corresponding two sides and included angle of the other triangle. In the figure, we have \(EF = QF\) (given by the marks on the sides) and \(FG=FG\) (common side).

Step2: Determine the additional known statement for part (a)

The additional known statement for part (a) is \(\angle EFG=\angle QFG\). The reason is that this angle is the included angle between the pairs of equal sides \(EF = QF\) and \(FG = FG\) which is required for the SAS congruence criterion.

Step3: Determine the information still needed for part (b)

If we assume we are trying to prove \(\triangle EFG\cong\triangle QFG\), we already have two sides (\(EF = QF\) and \(FG=FG\)). For SAS, we need the included angle. If the given was about another pair of triangles (but based on the standard figure interpretation), if there was a mis - identification of sides, we still need to confirm the equality of the included angles between the two pairs of sides. But since we already used the included angle for the standard pair \(\triangle EFG\) and \(\triangle QFG\) in part (a), if we consider a wrong initial assumption (but based on the problem structure), there is no other information. But if we follow the problem's flow, after stating the included angle in part (a), for a general case (if there was a miscalculation in side - angle pairing at first glance), we need to ensure that the sides and the included angle are correctly identified. But based on the problem's design:

Answer:

a) Additional Known Statement: \(\angle EFG=\angle QFG\); Reason: Included angle for SAS congruence.
b) Information still needed: (If we assume a wrong initial side - angle pairing at first (but not in the standard \(\triangle EFG\) and \(\triangle QFG\) case where part (a) gives the correct included angle), but based on problem structure) There is no additional information needed as part (a) provides the included angle for the two triangles \(\triangle EFG\) and \(\triangle QFG\) with \(EF = QF\), \(FG = FG\) (common side) and \(\angle EFG=\angle QFG\) (from part (a)) to satisfy SAS. But if we follow the problem's two - part structure strictly (as a teaching moment to check understanding of SAS): If we wrongly thought of another pair (but no), for the standard \(\triangle EFG\) and \(\triangle QFG\), after part (a) there is no more needed. But if we consider the problem's format (maybe a mis - print in problem numbering), but based on geometric principles: For proving two triangles congruent by SAS, we need two sides and the included angle. If in a wrong initial analysis (but corrected by part (a)), no. But as per the problem's two - part: a) \(\angle EFG=\angle QFG\), SAS (included angle); b) (Since part (a) gives the included angle) no (but if we consider a student's error in not seeing part (a) as the solution) but no. So final answer:
a) Additional Known Statement: \(\angle EFG = \angle QFG\); Reason: Included angle for SAS.
b) Information still needed: (If we assume a wrong start, but no) (Based on correct geometric approach with part (a) answering the SAS requirement) No additional information (as part (a) provides the included angle for the two triangles \(\triangle EFG\) and \(\triangle QFG\) with \(EF=QF\), \(FG = FG\) (common) and \(\angle EFG=\angle QFG\))