ABC is a triangle in which ∠B=2∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD. Prove that ∠BAC = 72°.
Answers
Given: ABC is a triangle in which ∠B = 2∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
To Prove : ∠BAC = 72°.
Proof :
Construct the angular bisector of ∠ABC, which meets AC in P and join PD.
Let, ∠ACB = y
∠B = ∠ABC = 2∠C = 2y
Let ∠BAD = ∠CAD = x
∠BAC = 2x
[ AD is the bisector of ∠BAC]
In ∆BPC, we have,
∠CBP = ∠BCP = y
[ BP is the bisector of ∠ABC)
Therefore, PC = BP
In ∆ABP & ∆DCP ,
∠ABP = ∠DCP = y
AB = DC (Given)
PC = BP (Proved above)
So, by SAS congruence criterion , we obtain ∆ABP ≅ ∆DCP
Now,
∠BAP = ∠CDP and, AP = DP (By CPCT)
∠BAP = ∠CDP = 2x and ∠ADP = ∠DAP = x
In ∆ABD,
∠ADC = ∠ABD + ∠BAD
[By exterior angle theroem ]
(∠ADP + ∠CPD) = ∠ABD + ∠BAD
x + 2x = 2y + x
x - x + 2x = 2y
2x = 2y
x = y …………(1)
In ∆ABC, we have,
We know that, the sum of all angles of triangle is 180°.
∠A + ∠B + ∠C = 180°
2x + 2y + y = 180°
[∠A = 2x, ∠B = 2y, ∠C = y]
2x + 2x + x = 180°
[From eq1 , x = y]
5x = 180°
x = 180°/5
x = 36°
∠BAC = 2x = 2 × 36°
∠BAC = 72°
Hence, ∠BAC = 72°.
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Answer:
Step-by-step explanation: Given: ABC is a triangle in which ∠B = 2∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
To Prove : ∠BAC = 72°.
Proof :
Construct the angular bisector of ∠ABC, which meets AC in P and join PD.
Let, ∠ACB = y
∠B = ∠ABC = 2∠C = 2y
Let ∠BAD = ∠CAD = x
∠BAC = 2x
[ AD is the bisector of ∠BAC]
In ∆BPC, we have,
∠CBP = ∠BCP = y
[ BP is the bisector of ∠ABC)
Therefore, PC = BP
In ∆ABP & ∆DCP ,
∠ABP = ∠DCP = y
AB = DC (Given)
PC = BP (Proved above)
So, by SAS congruence criterion , we obtain ∆ABP ≅ ∆DCP
Now,
∠BAP = ∠CDP and, AP = DP (By CPCT)
∠BAP = ∠CDP = 2x and ∠ADP = ∠DAP = x
In ∆ABD,
∠ADC = ∠ABD + ∠BAD
[By exterior angle theroem ]
(∠ADP + ∠CPD) = ∠ABD + ∠BAD
x + 2x = 2y + x
x - x + 2x = 2y
2x = 2y
x = y …………(1)
In ∆ABC, we have,
We know that, the sum of all angles of triangle is 180°.
∠A + ∠B + ∠C = 180°
2x + 2y + y = 180°
[∠A = 2x, ∠B = 2y, ∠C = y]
2x + 2x + x = 180°
[From eq1 , x = y]
5x = 180°
x = 180°/5
x = 36°
∠BAC = 2x = 2 × 36°
∠BAC = 72°
Hence, ∠BAC = 72°.
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