Q. 193.7( 3 Votes )
In Fig. 10.141, ABC is a triangle in which ∠B =2∠C. D is a point on side such that AD bisects ∠BAC and AB=CD. BE is the bisector of ∠B. The measure of ∠BAC is
[Hint: Δ ABE ≅ Δ DCE]
A. 72°
B. 73°
C. 74°
D. 95°
Answer :
Given that 
ABC
BE is bisector of ∠Band AD is bisector of ∠BAC
∠B = 2 ∠C
By exterior angle theorem in triangle ADC
∠ADB = ∠DAC + ∠C (i)
In 
ADB,
∠ABD + ∠BAD + ∠ADB = 180o
2 ∠C + ∠BAD + ∠DAC + ∠C = 180o [From (i)]
3 ∠C + ∠BAC = 180o
∠BAC = 180o – 3 ∠C (ii)
Therefore,
AB = CD
∠C = ∠DAC
∠C = 1/2 ∠BAC (iii)
Putting value of Angle C in (ii), we get
∠BAC = 180o – 1/2 ∠BAC
∠BAC + 
∠BAC = 180o
∠BAC = 180o
∠BAC = 
= 72o
∠BAC = 72o
Rate this question :
Champ Quiz | Mastering Triangles44 mins
RD Sharma | Important Proofs in Triangles28 mins
R.D. Sharma | Triangles33 mins
NCERT | Most Important Questions of Triangles50 mins
Quiz | Are You Master of Triangles?48 mins
NCERT | Tricks to Apply Triangle Inequalities45 mins
Triangles - Important Questions39 mins
RD Sharma | Problems on Triangles41 mins
NCERT | Triangle Inequalities47 mins
NCERT | Practice Imp. Qs. on Triangles44 mins
Prove that the angles opposite to equal sides of a triangle are equal
RS Aggarwal & V Aggarwal - MathematicsIf the sides of a triangle are produced in order, then the sum of the three exterior angles so formed is
RD Sharma - MathematicsIn Δ ABC, if ∠A = 100° AD bisects ∠A and AD⊥BC. Then, ∠B =
RD Sharma - MathematicsD is any point on side AC of a ΔABC with AB = AC. Show that CD < BD.
NCERT Mathematics ExemplarIn a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
RD Sharma - MathematicsIn Δ ABC, ∠A=50° and BC is produced to a point D. If the bisectors of ∠ABC and ∠ACD meet at E, then ∠E =
RD Sharma - MathematicsCompute the value of x in each of the following figures:
(i)
(ii)
(iii)
(iv)
In Δ PQR, if PQ=QR and L, M and N are the mid-point of the sides PQ, QR and RP respectively. Prove that LN=MN.
RD Sharma - MathematicsIn a right-angled triangle, one of the acute measures 53°. Find the measure of each angle of the triangle.
RS Aggarwal & V Aggarwal - MathematicsProve that the perimeter of a triangle is greater than the sum of its altitudes.
RD Sharma - Mathematics