The sides AB and AC of AABC are produced D and E, respectively. The bisectors of ∠CBD and ∠BCE meet at P. If ∠P = 54° find the measure of ∠A.

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  1. 72°
  2. 36°
  3. 48°
  4. 28°

Answer (Detailed Solution Below)

Option 1 : 72°
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Detailed Solution

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Given:

∠P = 54º

Sides AB and AC of ΔABC are produced to D and E, respectively.

The bisectors of ∠CBD and ∠BCE meet at P.

Formula Used:

Exterior Angle Property: ∠P = ∠CBD + ∠BCE

Sum of angles in a triangle: ∠A + ∠B + ∠C = 180º

Calculation:

Using the exterior angle property:

∠P = ∠CBD + ∠BCE

⇒ 54º = ∠B + ∠C

Using the sum of angles in a triangle:

∠A + ∠B + ∠C = 180º

⇒ ∠A + 54º = 180º

⇒ ∠A = 180º - 54º

⇒ ∠A = 72º

The measure of ∠A is 72º.

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