1. **Problem statement:** We are given a triangle with an incenter and need to find the measures of angles and segments related to the incenter and the triangle. 2. **Key concept:** The incenter of a triangle is the point where the angle bisectors meet, and it is the center of the inscribed circle (incircle) that touches all sides. 3. **Important rules:** - The incenter is equidistant from all sides of the triangle. - The angle bisectors split the angles into two equal parts. - The sum of angles in a triangle is always $180^\circ$. 4. **Step-by-step solution:** - Given angles: $\angle A = 30^\circ$, $\angle D = 67^\circ$ (at point D towards QB), and segments AP = 13, PB = 24, QR = 10. - Since D is the incenter, it lies on the angle bisectors. Therefore, angles at B and C can be found using the triangle angle sum: $$\angle B + \angle C + \angle A = 180^\circ$$ $$k + m + 30 = 180$$ $$k + m = 150$$ - The angle at D (67°) is related to the bisected angles, so we can use it to find $k$ and $m$ if more information is given. - The segments h, w, y are perpendicular distances from D to sides AB, BC, and AC respectively, and since D is the incenter, these distances are equal: $$h = w = y$$ - Using the given segment lengths and properties of the incircle, you can set up equations to solve for these variables. 5. **Summary:** - Use the angle sum property to find unknown angles. - Use the property of the incenter being equidistant from sides to find segment lengths. - Use angle bisector properties to relate angles at the incenter. Since the problem is complex and requires the diagram for exact numeric answers, this explanation guides you on how to approach it step-by-step.