Suppose that ∡BOC ≅ ∡DOE by the vertical angle theorem Angle AOB is congruent to angle DOE and Angle BOC is congruent to angle EOF by the transitive property, Angle EOF is not congruent to angle DOE and Angle AOB is not congruent to angle EOF which contradicts the given. Therefore ∡BOC ≅ ∡DOE
How do we prove that ∡BOC ≅ ∡DOE?
Suppose ∡BOC is not congruent to ∡DOE
By the vertical angle theorem, ∡BOC is congruent to ∡AOD, and ∡DOE is congruent to ∡COB. This is because opposite angles formed by intersecting lines are congruent.
Since ∡AOB is congruent to ∡EOF (given), and we have established that ∡AOD (which is the same as ∡BOC by vertical angles) is congruent to ∡BOC, then by the transitive property, ∡EOF should also be congruent to ∡BOC.
if ∡BOC is not congruent to ∡DOE, then ∡EOF should not be congruent to ∡DOE. This contradicts the established fact from step 3 that ∡EOF is congruent to ∡BOC.
Full question
Three lines BE, AD, and CF are intersected at the midpoint point O
Given: ∡AOB ≅ ∡ EOF
Prove: ∡BOC ≅ ∡DOE
Complete the proof
Suppose that ∡BOC ≅ ∡DOE by the vertical angle theorm .................. and ....................., by the transitive property, ................ and ................... which contradicts the given. Therefore ∡BOC ≅ ∡DOE ..................