If $\angle BOC=80^{\circ}$ and $OA$ bisects $\angle BAC$, then find the value of $\angle ABO$ in degrees.

✨ Free AI Tools for You

Powered by Chatterbot AI


Given: $\angle BOC = 80^\circ$ and $OA$ is a line that bisects $\angle BAC$.

We know that, an angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, we can write:

$$ \angle BOC = 2 \times \angle BAC $$

Substituting the given value:

$$ 80^\circ = 2 \times \angle BAC \ \therefore \angle BAC = 40^\circ $$

Since $OA$ bisects $\angle BAC$, we can conclude:

$$ \angle BAO = \frac{\angle BAC}{2} = \frac{40^\circ}{2} = 20^\circ $$

Now, consider the triangle $\triangle AOB$:

Since $OA = OB = r$ (radii of the circle), $\triangle AOB$ is isosceles, making the angles opposite the equal sides equal. Therefore:

$$ \angle OAB = \angle OBA $$

Given that $\angle BAO = 20^\circ$, we find:

$$ \angle ABO = 20^\circ $$

So, the value of $\angle ABO$ is 20 degrees.

Was this helpful?

India's 1st AI Doubt Solver for CBSE, JEE, and NEET

Ask a Question for Free

and then it's just ₹212 a month