Question 17

Explanation

We are asked to determine the measure of angle $\angle BOC$ in equilateral triangle $ABC$ with the intersection of lines $AP$, $BQ$, and $CR$ at point $O$.

Analyzing Initial Information

Triangle $ABC$ is an equilateral triangle. This means:

• $AB = BC = CA$
• $\angle A = \angle B = \angle C = 60^\circ$

However, we do not yet know the properties of lines $AP$, $BQ$, and $CR$ for certain from the initial problem, except that they intersect at $O$.

Analyzing Statement (1)

Statement (1): $BC = 10 \text{ cm}$.

This information only provides the length of the triangle's side. Without knowing the properties of lines $BQ$ and $CR$ (whether they are altitudes, angle bisectors, or medians), we cannot determine the position of point $O$ and the angles within it with certainty.

So, statement (1) ALONE is not sufficient.

Analyzing Statement (2)

Statement (2): Lines $AP$, $BQ$, and $CR$ are angle bisectors of triangle $ABC$.

If $BQ$ is the angle bisector of angle $B$, then:

If $CR$ is the angle bisector of angle $C$, then:

Now consider triangle $BOC$. The sum of angles in a triangle is $180^\circ$.

We can determine the measure of angle $\angle BOC$ using only statement (2).

So, statement (2) ALONE is sufficient.

Conclusion

Statement (1) is not sufficient, but statement (2) is sufficient.