Question

Question was provided via image upload.

Explanation:

Identify given geometric markings

From the image, we observe the following markings on the geometric figure:

1. Segment \(FG\) has a single tick mark, and segment \(HG\) has a single tick mark. This indicates that \(FG \cong HG\).
2. Line segments \(EF\) and \(HJ\) have matching parallel arrows, which indicates that \(EF \parallel HJ\).

Identify vertical angles

Using the Triangle Geometry concept, we identify that lines \(EJ\) and \(FH\) intersect at point \(G\). Therefore, \(\angle EGF\) and \(\angle HGJ\) are vertical angles, which means:

$$\angle EGF \cong \angle HGJ \quad \text{(Angle)}$$

Identify alternate interior angles

Since \(EF \parallel HJ\) and line \(FH\) acts as a transversal line intersecting them, the alternate interior angles are congruent:

$$\angle EFG \cong \angle JHG \quad \text{(Angle)}$$

Similarly, with line \(EJ\) acting as a transversal line intersecting the parallel lines, we have:

$$\angle FEG \cong \angle HJG \quad \text{(Angle)}$$

Apply AAS Congruence Theorem

We have established the following set of congruent parts between \(\Delta EFG\) and \(\Delta JHG\):

• \(\angle FEG \cong \angle HJG\) (Angle)
• \(\angle EGF \cong \angle HGJ\) (Angle)
• \(FG \cong HG\) (Non-included Side)

Since we have two angles and a non-included side congruent, we can prove \(\Delta EFG \cong \Delta JHG\) using the Angle-Angle-Side (AAS) congruence theorem.

Apply ASA Congruence Theorem

Alternatively, we can use the following set of congruent parts:

• \(\angle EFG \cong \angle JHG\) (Angle)
• \(FG \cong HG\) (Included Side)
• \(\angle EGF \cong \angle HGJ\) (Angle)

Since the congruent side is directly between the two congruent angles, we can prove \(\Delta EFG \cong \Delta JHG\) using the Angle-Side-Angle (ASA) congruence theorem.

Answer:

• (A) ASA (Angle-Side-Angle) (Correct answer)
• (B) AAS (Angle-Angle-Side) (Correct answer)
• (C) SSS (Side-Side-Side)
• (D) SAS (Side-Angle-Side)
• (E) HL (Hypotenuse-Leg)