Question

which congruence theorems can be used to prove δefg ≅ δjhg? select two options. options: hl, sas, sss image of triangles efg and jhg with g as intersection point, marked congruent segments

Explanation:

Step1: Analyze SSS Congruence

From the diagram, we can see that there are markings indicating equal lengths. For \(\triangle EFG\) and \(\triangle JHG\), we can identify three pairs of equal sides. The vertical angles at \(G\) are equal, but more importantly, the side markings suggest \(EF = JH\), \(FG=HG\), and \(EG = JG\) (since \(G\) is the midpoint or due to the segment markings). So by SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Step2: Analyze SAS Congruence

We know that vertical angles \(\angle EGF\) and \(\angle JGH\) are equal (vertical angles theorem). Also, from the segment markings, we can find two pairs of equal sides. For example, \(FG = HG\) and \(EG=JG\), and the included angle between these sides (\(\angle EGF\) and \(\angle JGH\)) is equal. So by 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, the triangles are congruent. HL (Hypotenuse - Leg) is used for right - angled triangles, and there is no indication that these triangles are right - angled, so HL is not applicable.

Answer:

A. SAS, C. SSS (assuming the options are labeled as HL, SAS, SSS with SAS and SSS being the correct ones)