Problem 13

Explanation

We are asked to determine the measure of angle $\angle BPC$ in an isosceles triangle $ABC$ with $AC = BC$.

Analysis of Statement 1

Given $\angle CAB = 40^\circ$. Since $\triangle ABC$ is isosceles with $AC = BC$, the base angles are equal:

We can calculate the vertex angle $\angle ACB$:

However, this information does not specify the position of point $P$ on side $AB$. Point $P$ can be anywhere along the segment $AB$. Consequently, the measure of $\angle BPC$ varies depending on the position of $P$. Therefore, statement $(1)$ ALONE is not sufficient to answer the question.

Analysis of Statement 2

Given $PC$ is the angle bisector of triangle $ABC$. In this context, $PC$ is the bisector of angle $C$ intersecting side $AB$ at point $P$.

In an isosceles triangle with $AC = BC$, the angle bisector drawn from the vertex (the angle between the equal sides) to the base has special properties. It is also:

1. An altitude (perpendicular to the base).
2. A median (bisects the base).

Since $PC$ is an altitude, $PC \perp AB$. Thus, the angle formed is a right angle:

We obtain a definite value for $\angle BPC$. Therefore, statement $(2)$ ALONE is sufficient to answer the question.

Conclusion

Statement $(1)$ is not sufficient, while statement $(2)$ is sufficient. Thus, statement $(2)$ ALONE is sufficient, but statement $(1)$ ALONE is not sufficient.