https://file.aiquickdraw.com/mathbot/file/1727014144983gbe1a0tj.jpg

The image shows a geometric problem where:

- \(DE \parallel QR\),
- \(AP\) and \(BP\) are angle bisectors of \(\angle EAB\) and \(\angle RBA\), respectively,
- We are required to find \(\angle APB\).

### Step-by-step Solution:

1. Since \(DE \parallel QR\), and lines \(AB\) intersect these parallel lines, we can apply properties of transversals and parallel lines to understand the angle relationships.
   
2. Let \(\angle EAB = \theta_1\) and \(\angle RBA = \theta_2\).
   
3. Since \(AP\) and \(BP\) are the bisectors of \(\angle EAB\) and \(\angle RBA\), the angles at \(P\) are half of \(\theta_1\) and \(\theta_2\) respectively:
   \[
   \angle PAB = \frac{\theta_1}{2}, \quad \angle PBA = \frac{\theta_2}{2}
   \]

4. Using the exterior angle theorem (as lines \(DE \parallel QR\)), we know that:
   \[
   \theta_1 + \theta_2 = 180^\circ
   \]
   
5. Now, we can calculate \(\angle APB\) by using the angle sum property in the triangle:
   \[
   \angle APB = 180^\circ - \left( \frac{\theta_1}{2} + \frac{\theta_2}{2} \right)
   \]
   Substituting \(\theta_1 + \theta_2 = 180^\circ\):
   \[
   \angle APB = 180^\circ - \frac{180^\circ}{2} = 90^\circ
   \]

Thus, \(\angle APB = 90^\circ\).

Let me know if you would like further clarifications, and here are some additional questions for practice:

1. If \(\theta_1 = 120^\circ\) and \(\theta_2 = 60^\circ\), what are the individual angles at \(P\)?
2. What other geometric properties could we derive from the parallel lines in the diagram?
3. If we change the orientation of the lines \(DE\) and \(QR\), will the result for \(\angle APB\) change?
4. How would the solution change if \(AP\) and \(BP\) were not bisectors?
5. Can we generalize this method for other configurations of parallel lines and angle bisectors?

**Tip:** Always use the parallel line theorem when you have parallel lines intersected by a transversal—it simplifies angle calculations!

If DE is parallel to QR, and AP and BP are the bisectors of angles EAB and RBA respectively, find angle APB.

Learn how to find angle APB in a geometric setup where DE is parallel to QR, and AP and BP are the angle bisectors of EAB and RBA. This problem applies the exterior angle theorem and angle sum property of triangles. Suitable for Grades 9-11.

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