For triangle tri the following facts are given:
segment an is perpendicular to segment ri
an = 6 cm
ri = 8 cm
ta = 3.3 cm
ni = 1.6 cm

in two column format prove that triangle tan is congruent to triangle tri

We can see that the two triangles share an angle and all their sides are parallel, thus by Tales theorem, they are in Tales position, which proves that they are congruent.

we can see from figure that:
call x and y the unknown segments TA and TN, then we have:

3.3/(3.3 + x) = 6/8

8*3.3 = 6(3.3 + x)
x = 1.1

and then the proportion still holds for those sides as well as AN and RI

we also have:
y/(y + 1.6) = 6/8
8y = 6y + 9.6

y = 4.8

and then the proportion still holds for those sides as well as AN and RI

-2x + 7 = 7

subtract 7 from both sides.

-2x = 0

divide both sides by -2.

x = 0

the value of x is equal to 0.