Question

50 points!!!
Given
ZBAC = 2x + 10
ZBDC = 4x-4

Solve For
X=
mLBAC=
mBC=

Answer 1

Answer:

x = 7

$\angle BAC=24^{\circ}$

$m\angle \stackrel{\frown}{BC}=48^{\circ}$

Step-by-step explanation:

(Refer to the attachments)

By angles subtended by same arc theorem,

$\angle BAC=\angle BDC$

$2x + 1 0 = 4x - 4$

$\implies1 0 + 4= 4x - 2x$

$\implies1 4= 2x$

$\implies \boxed{x = 7}$

Given that $m \angle BAC=2x+10=2*7+10=24^{\circ}$

By the Angle at the center theorem

• $m\angle \stackrel{\frown}{BC}=2*24=48^{\circ}$

Answer 2

Answer:

$\angle BAC = \angle BDC$

$x = 7$

$\sf m \angle BAC = 24^\circ$

$\sf m \angle BDC = 24^\circ$

$\sf m\angle \stackrel{\frown}{BC}= 48^\circ$

Step-by-step explanation:

We know that:

The inscribed angle subtended by the same arc is equal.

Using this, we can say that:

$\sf \angle BAC = \angle BDC$

Substitute the value:

$2x + 10 = 4x - 4$

Let's solve for $x$:

$2x - 4x = -4 - 10$

$-2x = -14$

$x =\dfrac{-14}{-2}$

$x = 7$

Now that we have found $x = 7$, we can substitute it back into the equation to find the values of the angles.

$\sf m \angle BDC = 2x + 10 = 2(7) + 10 = 14 + 10 = 24^\circ$

$\sf m \angle BAC = 4x - 4 = 4(7) - 4 = 28 - 4 = 24^\circ$

And,

The Inscribed Angle Theorem states that an angle inscribed in a circle is equal to half the measure of the intercepted arc.

So,

$\sf m\angle BAC = \dfrac{1}{2} \cdot m\angle \stackrel{\frown}{BC}$

$\sf m\angle \stackrel{\frown}{BC}= 2 \cdot m\angle BAC$

Substitute the value:

$\sf m\angle \stackrel{\frown}{BC}= 2 \cdot 24^\circ$

$\sf m\angle \stackrel{\frown}{BC}= 48^\circ$

Therefore, the answers are:

$\angle BAC = \angle BDC$

$x = 7$

$\sf m \angle BAC = 24^\circ$

$\sf m \angle BDC = 24^\circ$

$\sf m\angle \stackrel{\frown}{BC}= 48^\circ$