$A B C$ is a triangle in which $∠ B = 2 ∠ C$ . $D$ is point on the side $B C$ such that $A D$ bisects $∠ B A C$ and $A B = C D$ . Prove that $∠ B A C = 72^{o}$ .

Open in App

Solution

$∠ B A P = ∠ C D P = 2 x$
$A P = D P$
$△ A D P, A P = D P$
$\Rightarrow ∠ A D P = ∠ D A P = x$
$∠ B = 2 ∠ C △ B P C$
$A B = C D ∠ C B P = a ∠ B C P = y$
$∠ B A C = 72^{o} \Rightarrow B P = P C$
$△ A B P$ and $△ D C P$
$∠ A B P = ∠ D C P = y$
$A B = D C$ and $B P = P C$
$△ A B P ≅ △ D C P$
in $△ A B D$ , $∠ A D C = ∠ A B D + ∠ B A D$
$\Rightarrow 3 x = 2 y + x$
$x = y$
in $△ A B C$ , we have $∠ A + ∠ B + ∠ C = 180^{o}$
$2 x + 2 y + y = 180^{o}$
$5 x = 180^{o}$
$x = 36$
$∠ B A C = 2 x = 72^{o}$

Suggest Corrections

Similar questions

Q. 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°.

Q. ABC is a triangle in which

∠ B = 2 ∠ C

. D is a point on BC such that AD bisects

∠ B A C

and

A B = C D .

Find

∠ B A C

Q. ABC is a triangle in which

∠ B = 2 ∠ C .

D is a point on side BC such that AD bisects

∠ B A C

and AB =CD. Find

m

∠ B A C = m^{\circ}

In the figure, ABC is a triangle in which $∠ B = 2 ∠ C . D is a point on side BC such that AD bisects ∠ B A C a n d A B = C D . B E is the bisector of ∠ B . The measure of ∠ B A C i s$

Q. D is a point on the side BC of a

△ A B C

such that AD bisects

∠ B A C

. Then

Join BYJU'S Learning Program

ABC is a triangle in which ∠B=2∠C. D is point on the side BC such that AD bisects ∠BAC and AB=CD. Prove that ∠BAC=72o.

∠BAP=∠CDP=2xAP=DP△ADP,AP=DP⇒∠ADP=∠DAP=x∠B=2∠C△BPCAB=CD ∠CBP=a∠BCP=y∠BAC=72o⇒BP=PC△ABP and △DCP∠ABP=∠DCP=yAB=DC and BP=PC△ABP≅△DCPin △ABD, ∠ADC=∠ABD+∠BAD⇒3x=2y+xx=yin △ABC, we have ∠A+∠B+∠C=180o 2x+2y+y=180o 5x=180o x=36∠BAC=2x=72o

$A B C$ is a triangle in which $∠ B = 2 ∠ C$ . $D$ is point on the side $B C$ such that $A D$ bisects $∠ B A C$ and $A B = C D$ . Prove that $∠ B A C = 72^{o}$ .