I'm not sure how to do part b. I know how to do part c. It would be best if you could present me with a diagram, as I already have my own diagram drawn.

Question

520.Given triangle ABC, let F be the point where segment BC meets the bisector of angle BAC. Draw the line through B that is parallel to segment AF, and let E be the point where this parallel meets the extension of segment CA.

(a) Find the four congruent angles in your diagram.

(b) How are the lengths EA, AC, BF, and FC related?

(c) The Angle-Bisector Theorem: How are the lengths AB, AC, BF, and FC related?

Answer 1

sorry it's not easy to put diagrams here

this problem steps through how to prove the angle bisector theorem

BE parallel to AF means angle EBC=FAC,,,,opposite alternate angle when AB is intersecting line,,,

and angle BEC=FAC ,,,,parallels create same angle along EA

which means triangle BEA is isosceles (later will use AE=AB since isosceles)

corrected typo

two triangles similar AFC and EBC

[(EA+AC)/(BF+CF)]=AC/CF

use algebra or always know this means

(AC/CF)=(EA/BF) which is same as (CF/BF)=(AC/AE)

substitute AB for AE

(CF/BF)=(AC/AB)

note: the 4th congruent angle is BAF since angle BAC is bisected.