Question

complete the proof that ( \triangle ehi cong \triangle fgi ).

Explanation:

Step1: Markings on the figure

From the figure, the markings show that \( \overline{EI}\cong\overline{FI}\) (given by the two - tick marks on the segments) and \( \overline{HI}\cong\overline{GI}\) (given by the single - tick marks on the segments).

Step2: Vertical angles

\(\angle EIH\) and \(\angle FIG\) are vertical angles. By the vertical angles theorem, vertical angles are congruent. So, \(\angle EIH\cong\angle FIG\).

Step3: SAS congruence criterion

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In \(\triangle EHI\) and \(\triangle FGI\), we have \(EI = FI\), \(HI=GI\), and \(\angle EIH=\angle FIG\).

Answer:

1. Reason: Given (from the segment markings in the figure)
2. Reason: Given (from the segment markings in the figure)
3. Reason: Vertical Angles Theorem
4. Reason: Side - Angle - Side (SAS) Congruence Criterion