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

Question

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

 

Answer 1

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