in the fig cd is the angle bisector of angle c , angle b = angle ace . Prove that angle adc  =  angle acd

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djchamp007

14.02.2018

Math

Secondary School

+6?pts

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In the figure CD is the angle bisector of angle C, angle B = angle ACE. Prove that angle ADC = angle ACD

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deepanshusingh1?

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Helping Hand

Ad is equal to ac (isoceles triangle's equal sides)
So if sides of isosceles triangle are equal so angles are also equal so angle ADC is equal to angle ACD
Hence proved

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bhagyashreechowdhury?

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Ace

Given:

CD is the bisector of angle C

Angle B = Angle ACE

To find:

Prove that angle ADC = angle ACD

Solution:

In ?BCD, we have

?BCD + ?BDC + ?DBC = 180? ....... [Angle sum property] .... (i)

Also, ?ADC + ?BDC = 180? ..... [Linear Pair] ...... (ii)

From equation (i) & (ii), we get

?BCD + ?BDC + ?DBC = ?ADC + ?BDC

? ?BCD + ?BDC + ?DBC - ?BDC = ?ADC

???BCD + ?DBC = ?ADC?............... (iii)

Now, according to the question we have,

?ABC = ?ACE ...... (iv)

and

?BCD = ?ECD ?.... [? CD is the bisector of angle C as given in the figure]...(v)

On adding eq. (iv) & (v), we get

?ABC + ?BCD = ?ACE + ?ECD

???ABC + ?BCD = ?ACD?...... (vi)

On comparing eq. (iii) & (vi), we get

??ADC = ??ACD

Hence proved