abc is a triangle with angle b = 2(angle c) d is a point on bc such that ad - Maths - Triangles

Question

abc is a triangle with angle b = 2(angle c) d is a point on bc
such that ad bisects angle bac and ab = cd Prove angle bac = 72
degree

Aakash Sharma · Accepted Answer

Dear student, The solution to question 4 is given below: In ΔABC, we have ∠B = 2∠C or, ∠B = 2y, where ∠C = y AD is the bisector of ∠BAC So, let ∠BAD = ∠CAD = x Let BP be the bisector of ∠ABC Join PD In ΔBPC, we have ∠CBP = ∠BCP = y ⇒ BP = PC (1)

Now, in ΔABP and ΔDCP, we have ∠ABP = ∠DCP = y AB = DC [Given] and, BP = PC [Using (1)] So, by SAS congruence criterion, we have $Δ ABP ≅ Δ DCP \Rightarrow ∠ BAP = ∠ CDP and AP = DP \Rightarrow ∠ CDP = 2 x ∠ ADP = DAP = x [∴ ∠ A = 2 x]$ In ΔABD, we have ∠ADC = ∠ABD + BAD ⇒ x + 2x = 2y + x ⇒ x = y In ΔABC, we have ∠A + ∠B + ∠C = 180° ⇒ 2x + 2y + y = 180° ⇒ 5x = 180° ⇒ x = 36° Hence, ∠BAC = 2x = 72° Hope this information will clear your doubts about the topic For remaining queries, we request you to post them in separate threads to have rapid assistance from our experts Regards

abc is a triangle with angle b = 2(angle c). d is a point on bc such that ad bisects angle bac and ab = cd.Prove angle bac = 72 degree