In triangle ABC, points D, E and F are on the sides AB, BC and AC respectively such that ED is angle bisector of AEB and EF is angle bisector of AEC. Further angle BDE = angle ADF and angle EFC = angle AFD. Find the measure of angle A.
Answers
Given : In triangle ABC, points D, E and F are on the sides AB, BC and AC respectively such that ED is angle bisector of AEB and EF is angle bisector of AEC.
∠BDE =∠ADF and ∠EFC =∠AFD
To find : measure of angle A.
Solution:
Refer attached figure
ED is angle bisector of AEB
∠BED = ∠AED = x
EF is angle bisector of AEC
∠CEF = ∠AEF = y
x + x + y + y = 180° straight line
=> x + y = 90°
∠BDE =∠ADF = m
∠EFC =∠AFD = n
in Δ BDE
x + m + ∠B = 180°
in Δ CFE
y + n + ∠C = 180°
Adding both
x + y + m + n + ∠B + ∠C = 180° + 180°
=> 90° + m + n + ∠B + ∠C = 360°
=> m + n + ∠B + ∠C = 270°
in ΔABC
∠A + ∠B + ∠C = 180° => ∠B + ∠C = 180° - ∠A
in ΔADF
∠A + m + m = 180° => m + n = 180° - ∠A
=> 180° - ∠A + 180° - ∠A = 270°
=> 2∠A = 90°
=> ∠A = 45°
measure of angle A. = 45°
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