Question

In the given triangle ABC, BC is produced to D. BF and FC are the bisectors of ∠ABC and ∠ACD respectively. If ∠BAC = 70° and EF = EC, then the measure of ∠FEC is equal to

• A. 80°
• B. 55°
• C. 110°
• D. 75°

Answer 1

The correct option is C 110°


Since BF is the angle bisector of ∠ABC.
$\therefore \angle FBC=\angle ABF=\frac{1}{2}\angle ABC$ ....(i)
Similarly, CF is the angle bisector of ∠ACD.
$\therefore \angle FCD=\angle ACF=\frac{1}{2}\angle ACD$ ....(ii)
Now, in ΔBFC,
∠BFC + ∠FBC = ∠FCD (Exterior angle property)
⇒ ∠BFC = ∠FCD – ∠FBC …..(iii)
Now, in ΔABC,
∠BAC + ∠ABC = ∠ACD (Exterior angle property)
⇒ 70° + 2∠FBC = 2∠FCD [From (i) and (ii)]
⇒ 70° = 2(∠FCD – ∠FBC)
⇒ 2∠BFC = 70° [From (iii)]
⇒ ∠BFC = 35° …..(iv)
In ΔECF, EC = EF (Given)
Now, angles opposite to equal sides are equal.
∴ ∠ECF = ∠EFC
⇒ ∠ECF = 35° [From (iv)] …..(v)
In ΔECF, by Angle sum property,
∠FEC + ∠ECF + ∠EFC = 180°
⇒ ∠FEC + 35° + 35° = 180° [From (iv) and (v)]
⇒ ∠FEC = 180° – 70°
⇒ ∠FEC = 110°
Hence, the correct answer is option (c).