In Δ ABC, BD ⊥ AC at D and ∠DBC = 54º. E is a point on BC such that ∠CAE= 34º. What is the measure of ∠AEB?

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  1. 80º
  2. 56º
  3. 78º
  4. 70º

Answer (Detailed Solution Below)

Option 4 : 70º
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Detailed Solution

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Given:

In Δ ABC, BD ⊥ AC at D and ∠DBC = 54º. E is a point on BC such that ∠CAE = 34º.

Formula Used:

Sum of angles in a Δ = 180º

Calculation:

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In Δ BDC, since BD ⊥ AC:

∠BDC = 90º

We have, ∠DBC = 54º

Then, ∠BDC + ∠DCB + ∠DBC = 180º

⇒ 90º + ∠DCB + 54º = 180º

DCB = 180º - 144º

DCB = 36º

In Δ AEC, ∠CAE = 34º

Then, ∠AEB = ∠CAE + DCB

⇒ ∠AEB = 34º + 36º

⇒ ∠AEB = 70º

Correct option is 70° .

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