Point A is the incenter of (triangle) PQR. Find each measure.

ARU

AU

QPK

The measures of ARU, AU, and QPK are all equal to the radius of the incircle of triangle PQR.

To find the measures of the angles ARU, AU, and QPK, we need to use the properties of the incenter of a triangle.

The incenter of a triangle is the point where the angle bisectors of the triangle intersect. In other words, it is the center of the inscribed circle within the triangle.

To find the measure of each angle, we will use the following properties:

1. The angles formed between the incenter and the vertices of the triangle are equal.
2. The sum of the measures of the three angles formed by the incenter and two vertices is always 180 degrees.

Let's break it down:

1. Angle ARU:
Since A is the incenter of triangle PQR, the angle formed by the incenter and any two vertices (in this case, A, R, U) will be equal. So the measure of angle ARU is equal to the measure of angle AUR.

2. Angle AU:
Similarly, the measure of angle AU is equal to the measure of angle AR.

3. Angle QPK:
Again, the measure of angle QPK is equal to the measure of angle QKP.

Now, to find the specific measures, we will need additional information such as the measures of angles P, Q, and R or any other relevant information about the triangle. Without that specific information, we cannot provide the numerical measures of these angles.

To find the measures, we need more information. Please provide the necessary angles or distances for each measure.