Question

ABC is triangle. AD is Perpendicular to BC. BE is perpendicular to AC. BE and AD cuts on F. BF =AC. find angle ABC
​

Answer 1

Answer:

∠ABC = 45°

Step-by-step explanation:

BC is triangle. AD is Perpendicular to BC. BE is perpendicular to AC. BE and AD cuts on F. BF =AC. find angle ABC

​

Let say ∠BFD = α

Then ∠DBF = 90-α   ( as AD ⊥ BC)

∠EFA = ∠BFD = α  ( opposite angles)

∠EAF = 90-α  ( as BE ⊥ AC)

∠EAF = ∠CAD  = 90-α ( as E lies on AC & F lies on AD)

∠ACD = 90 - (90-α) = α   ( as AD ⊥ BC)

Now comparing Triangles

Δ BDF  & Δ ADC

∠BDF = ∠ADC = 90°

∠BFD = ∠ACD = α

∠DBF = ∠CAD  = 90-α

Hence Δ BDF  ≅ Δ ADC

BF/AC = BD/AD

BF = AC ( given)

=> BD = AD

now in Δ ABD  , BD = AD

=>∠ABD = ∠BAD = β

∠ABD + ∠BAD + ∠ADB = 180°

=>  β + β + 90° = 180°

=> 2β = 90°

=> β = 45°

∠ABD  = 45°

∠ABC = ∠ABD  ( as D lies lies on BC)

=>∠ABC = 45°