Question

1. given: ab ≅ cg, ab || cg prove: △bca ≅ △gac statements reasons

Explanation:

Step1: Identify the included angle

Since \(AB\parallel CG\), we can use the property of parallel lines. The alternate - interior angles are equal. So, \(\angle BAC=\angle GCA\) (Alternate - interior angles formed by parallel lines \(AB\) and \(CG\) and transversal \(AC\)).

Step2: Use the SAS (Side - Angle - Side) congruence criterion

We are given that \(AB = CG\) (given). Also, \(AC=CA\) (common side). And we have just found that \(\angle BAC=\angle GCA\).
By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

Statements	Reasons
\(AB\parallel CG\)	Given
\(\angle BAC=\angle GCA\)	Alternate - interior angles (parallel lines \(AB\) and \(CG\) with transversal \(AC\))
\(AC = CA\)	Reflexive property of equality
\(\triangle BCA\cong\triangle GAC\)	SAS (Side - Angle - Side) congruence criterion (\(AB = CG\), \(\angle BAC=\angle GCA\), \(AC = CA\))