To fill the table:
From the figure, we observe that,
First box:
[tex]HV\perp GT,GH\cong TV\text{ (Reason is, Given)}[/tex]
Second box:
[tex]\Delta IGH\text{ and }\Delta ITV\text{ are right (Reason is, Definition of perpendicular lines)}[/tex]
Third box:
[tex]I\text{ is the midpoint of HV (Reason is, Given)}[/tex]
Fourth box:
[tex]HI\cong IV(Definition\text{ of midpoint)}[/tex]
Fifth box:
[tex]\Delta\text{IGH}\cong\Delta\text{ITV (HL theorem)}[/tex]
