Abc is a triangle in which angle b = 2 angle c. d is a point on side bc such that ad bisects angle bac and ab = cd. prove that angle bac equal to 72 degrees

Step-by-step explanation:

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.

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

Therefore

∠BAP = ∠ CDP = 2x and AP = DP ,

So in Δ APD, AP=DP

=> ∠ADP = ∠DAP = x

In ΔABD, we have

∠ADC = ∠ABD + BAD ⇒ 3x= 2y + x

⇒ x = y

In ΔABC, we have

∠A + ∠B + ∠C = 180°

⇒ 2x + 2y + y = 180°

⇒ 5x = 180°

⇒ x = 36°

Hence, ∠BAC = 2x = 72°

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for