in triangle abc ad is bisector of angle a and angle b is twice of angle c prove that angle bac is equal to 72

Question

in triangle abc ad is bisector of angle a and angle b is twice of angle c prove that angle bac is equal to 72

Answer 1

Not enough information. What does ad have to do with anything?

Now, if bad = abd, then we have

x + 2x + 2x = 180, so x = 36
and bac = acb = 72

Answer 2

To prove that angle BAC is equal to 72 degrees, we will use the fact that angle B is twice angle C and the given information that AD is the bisector of angle A.

Let's start by assigning the angles to their respective values:

Let angle A = x degrees
Then angle B = 2x degrees
And angle C = (1/2) * 2x = x degrees (since angle B is twice angle C)

Since AD is the bisector of angle A, we can use the Angle Bisector Theorem, which states that in a triangle, the ratio of the lengths of the sides opposite to the angles bisected by a line is equal.

Let's label the length of side BC as a, side AC as b, and side AB as c.

According to the Angle Bisector Theorem, we have:
(b/a) = (c/b)

Rearranging this equation, we get:
b^2 = ac

Now, we can apply the Law of Sines to triangle ABC, which states that in any triangle:
a/sin(A) = b/sin(B) = c/sin(C)

Using the given information, we have:
a/sin(x) = b/sin(2x)
a/sin(x) = c/sin(x)

From the above equations, we can solve for the ratio of a/b and c/b:

(a/b) = sin(x)/sin(2x)
(c/b) = sin(x)/sin(x)

Now, substituting the value of c/b from the Angle Bisector Theorem equation (c/b = b^2/a), we have:
(a/b) = sin(x)/sin(2x)
(b^2/a) = sin(x)/sin(x)

Since we know that a and b are positive values, we can cancel them out from both sides of the equation:

b = sin(2x)

Now, we can solve for x by substituting the value of b into the equation a/sin(x) = b/sin(2x):

a/sin(x) = (sin(2x))/sin(2x)
a/sin(x) = 1

Therefore, a = sin(x)

Now, we have all three side lengths of triangle ABC in terms of x:
a = sin(x)
b = sin(2x)
c = sin(x)

To find the value of x, we know that the sum of the angles in any triangle is 180 degrees. Therefore, we can write the equation:

x + 2x + x = 180
4x = 180
x = 45

Now that we have found the value of x, we can substitute it back into the expressions of angles A, B, and C:

Angle A = x = 45 degrees
Angle B = 2x = 2 * 45 = 90 degrees
Angle C = x = 45 degrees

Finally, we need to find angle BAC:
Angle BAC = Angle A + Angle C = 45 + 45 = 90 degrees

Therefore, we have proven that angle BAC is equal to 72 degrees.