Question

if AE is a bisector of angle BAC and a triangle ABC and AD perpendicular BC then prove that angle DAE is equal to 1/2(angle B- angle C

Answer 1

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Answer 2

Answer:

∠DAE = 1/2(∠C − ∠B)

Step-by-step explanation:

Given:-

In ∆ABC;

AE is a bisector of ∠BAC

AD perpendicular BC

To Prove:-

∠DAE = 1/2(∠C − ∠B)

Proof:-

In ∆ABC, since AE bisects ∠A, then

∠BAE = ∠CAE                                        (i)

In ∆ADC,

∠ADC+∠DAC+∠ACD = 180°                              [∵Angle sum property]

⇒90° + ∠DAC + ∠C = 180°

⇒∠C = 90°−∠DAC                                       (ii)

Now,

In ∆ADB,

∠ADB+∠DAB+∠ABD = 180°                                   [∵Angle sum property]

⇒90° + ∠DAB + ∠B = 180°

⇒∠B = 180°-90°−∠DAB                                         (iii)

⇒∠B = 90°−∠DAB

Subtracting (iii) from (ii), we get;

∠C − ∠B =∠DAB − ∠DAC

⇒∠C − ∠B =[∠BAE+∠DAE] − [∠CAE−∠DAE]

⇒∠C − ∠B =∠BAE+∠DAE − ∠BAE+∠DAE                   [∵∠BAE = ∠CAE ]

⇒∠C − ∠B =  2∠DAE

⇒  ∠DAE =   1/2(∠C − ∠B)

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