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in triangle abc AB=AC , angle bisectors of angle B AND C intersect AC atE and AB at F respectively . prove that BF=FE=EC

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1 Answer

Statement.       |.          Reason
1. AB = AC.        1. Given
2. Angle B =       2. When 2 sides of a triangle are congruent, the angles opposite them are congruent 
    Angle C
3. AE bisects       3. Given
   angle A, BF bi-
   sects angle B
4. Angle EBF        4. Definition of an angle bisector
Equals angle 
EBC, Angle ECF
Equals angle FCB
5. Angle EBC =    5. Halves of equals are equals
 Angle FCB
6. BC = BC.         6. Identity property
7. Triangle EBC=  7. ASA = ASA
Triangle FCB
8. BF = EC          8. Corresponding parts of congruent triangles are congruent
9. AF =AE.          9. Subtraction Postulate
10. Angle AFE =.  10. Same as reason 2
Angle AEF
11. Angle A =.     11. Identity property
Angle A
12. Angle B +.     12. Subtraction Postulate.....Alltriangles interior angles add up to 180
Angle C = Angle
AEF + Angle AFE
13. Angle B =      13. Transitive property
Angle AFE
14. FE || BC.        14. If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent
15. Angle FEB =   15. If 2 parallel lines are cut by a transversal, then the alternate interior angles are equal
Angle CBE
16. Angle FEB=.    16. Transitive Property
Angle EBF