Prove that if lines a, b,c form a pencil, then there is a line d in the pencil such that ? () = ? (d).and its converse proof as well.

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.

yes

step-by-step explanation:

they have to be opposite and reciprocal of eachother.

c

step-by-step explanation: easy math