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In triangle ABC, AB = AC. D is a point on BC such that AB = CD. E on AB such that DE ⊥ AB. Prove that 2 ∠ADE = 3∠B.

In triangle ABC, AB = AC. D is a point on BC such that AB = CD. E on AB such that DE ⊥ AB. Prove that 2 ∠ADE = 3∠B.

Grade:10

1 Answers

Ram Kushwah
108 Points
4 months ago
Given that AB = AC and AB=DC
so in Δ ADC; ∠ DAC = ∠ ADC =y (say)
and let ∠ ADE=x
Here In Δ ADC
y+y+∠C=180
So y= 90 - ∠ C/2.............................(1)
Thus at point D
In the Δ BDE
∠B+∠BDE=90
∠BDE=90 - ∠B................................(2)
At point D the sum of three angles will be 180°
∠BDE +x+y=180
Putting value of ∠BDE from (2) we get
 
90 - ∠B+x+y=180
x+ y- ∠B=90
Putting value of y from (1)
x + 90 - ∠ C/2 - ∠B =90
x=∠ C/2+∠B
But ∠ C= ∠B
So x= ∠ B/2+∠B=3∠B/2
 
Thus 2x==3∠B
 
OR 3∠B=2∠ADE
 
 

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