So I kind of get proofing but I am lost for what you call it when two sides that aren't adjacent are supplementary. Is it just supplementary angles? I want to say either <3 and <1 are congruent or <2 and <3 are congruent. If I could get help on the rest of the problem that would be much appreciated.

Let angle 4 denote the angle adjacent to angle 3 on the right side (so between lines a and b, to the right of line c). Angles 3 and 4 are then supplementary.

Lines a and b are parallel, so angles 1 and 4 are congruent because they are alternating interior angles.

By the transitive property, angles 2 and 4 are also congruent.

Again, transitively, angles 2 and 3 are supplementary.

Let angle 5 denote the angle to the left of angle 2 (between a and b, to the left of d). Angles 5 and 2 are supplementary.

Transitively, angles 5 and 3 are congruent.

These two angles correspond to one another, so lines c and d are parallel.

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.