given a triangle ABC, and a point P inside the triangle such that angle PAC is 40, angle PAB is 30, angle PCA is 20 and angle PCB is 10, what is the value of angle PBA

Given the conditions in the triangle \( ABC \) with point \( P \) inside the triangle, we need to find the measure of angle \( \angle PBA \).

### Problem Analysis:
- \(\angle PAC = 40^\circ\)
- \(\angle PAB = 30^\circ\)
- \(\angle PCA = 20^\circ\)
- \(\angle PCB = 10^\circ\)

### Solution:
Let \( x = \angle PBA \), which is what we need to find.

**Step 1: Use the sum of angles around point \( P \)**

The angles \( \angle PAC \), \( \angle PAB \), \( \angle PCA \), and \( \angle PCB \) are connected to the angles at point \( P \). The sum of angles around point \( P \) inside the triangle is \( 360^\circ \). This leads us to consider the sum of angles in smaller parts of triangle ABC that include point P.

**Step 2: Finding angle \( \angle APC \)**

\[
\angle APC = 180^\circ - (\angle PAC + \angle PCA) = 180^\circ - (40^\circ + 20^\circ) = 120^\circ
\]

**Step 3: Finding angle \( \angle APB \)**

\[
\angle APB = 180^\circ - (\angle PAB + \angle PBA) = 180^\circ - (30^\circ + x)
\]

**Step 4: Using triangle sum for triangle PBC**

In triangle \( PBC \), the sum of internal angles is:

\[
\angle PBC + \angle PCB + \angle APB = 180^\circ
\]

Substituting the known values:

\[
x + 10^\circ + (150^\circ - x) = 180^\circ
\]

Simplify the equation:

\[
120^\circ = x 
\]

Hence the measure is x = 120^\circ.

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Given a triangle ABC, and a point P inside the triangle such that angle PAC is 40°, angle PAB is 30°, angle PCA is 20°, and angle PCB is 10°, what is the value of angle PBA?

Explore how to find the value of angle PBA in a triangle ABC with an internal point P, using angle sum rules and geometric theorems. This detailed solution is suitable for high school students in Grades 9-12, covering internal angle properties and the triangle angle sum theorem.

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