Internal and External Angle Bisectors of a Triangle

Grades 9-10
Student math problem: Internal and External Angle Bisectors of a Triangle
Original question
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Question: solve
To find the value of \angle PBC, we can use the properties of internal and external angle bisectors in a triangle.

1. Identifying the Geometry

In the given diagram, we are looking at \triangle ABC:
  • AP is the internal angle bisector of \angle BAC. This means \angle PAC = \frac{1}{2} \angle BAC.
  • The line AC is extended to D, making \angle BCD an exterior angle of the triangle at vertex C.
  • The text states " CD is the angle bisector of \angle BCD," which is likely a typo for CP (the ray shown in the diagram). We assume CP is the external angle bisector of \angle BCA. This means \angle PCD = \frac{1}{2} \angle BCD.

2. Finding the value of \angle ABC

There is a known geometric theorem stating that the angle formed by the intersection of an internal bisector of one angle and the external bisector of another angle in a triangle is equal to half of the third internal angle. Mathematically, for \triangle ABC:
\angle APC = \frac{1}{2} \angle ABC
Given that \angle APC = 25^\circ: 25^\circ = \frac{1}{2} \angle ABC \angle ABC = 50^\circ

3. Finding the value of \angle PBC

The point P is the intersection of an internal bisector ( AP) and an external bisector ( CP). In geometry, this point is called an excenter of the triangle. Specifically, since it lies on the internal bisector of A and the external bisector of C, it is the excenter opposite to vertex A.
A property of an excenter is that the line connecting the third vertex ( B) to the excenter ( P) must also be an angle bisector. Since P is the excenter opposite A, the line BP must be the external angle bisector of \angle ABC.
  1. First, we find the exterior angle at vertex B: \text{Exterior } \angle B = 180^\circ - \angle ABC = 180^\circ - 50^\circ = 130^\circ
  2. Since BP bisects this exterior angle, \angle PBC is half of 130^\circ: \angle PBC = \frac{130^\circ}{2} = 65^\circ
Therefore, the value of \angle PBC is 65^\circ.
geometryangle-bisectorstriangle-properties