Let

AB and
CD be two lines intersecting at point O. Let
OP be the bisector of
angle AOC,
OT the bisector of angle POB, and
OR the bisector of angle TOD. If angle POR = 25 degrees�,
find angle AOC and angle AOD.

To find the angles AOC and AOD, we need to use the given information about the bisectors and the known angle POR.

1. Let's start by finding the angle AOC:
Since OP is the bisector of angle AOC, we know that angle AOP is equal to angle POC. Let's denote this angle as x.

2. Since OT is the bisector of angle POB, we know that angle TOP is equal to angle BOP. Let's denote this angle as y.

3. Similarly, since OR is the bisector of angle TOD, we know that angle TOR is equal to angle DOR. Let's denote this angle as z.

Now, let's analyze the given information:

4. We know that angle POR is 25 degrees. We can see that it can be expressed as the sum of angles AOP, TOP, and TOR:
POR = AOP + TOP + TOR

5. Substituting the corresponding angles (x, y, and z), we have the equation:
25 = x + y + z

6. Since angle AOC is equal to the sum of angles AOP and TOP, we can express it as:
AOC = 2x + 2y

7. And angle AOD is equal to the sum of angles AOP, TOP, and TOR, which can be expressed as:
AOD = 2x + 2y + 2z

Now, let's solve the equation:

8. From the equation (5), we have:
x + y + z = 25

9. We can use this equation to find the values of x, y, and z.

10. Let's assume that x = 5. Then, substituting this into the equation (8), we have:
5 + y + z = 25

11. Simplifying the equation, we find:
y + z = 20

12. Now, let's find an acceptable combination of y and z that satisfies this equation.

13. From the equation, we can see that the possible values of (y, z) are (10, 10), (9, 11), (11, 9), (8, 12), (12, 8), etc.

14. For each valid combination of y and z, we can calculate the corresponding values of x, AOC, and AOD.

15. For example, if we choose (y, z) = (10, 10):
x = 5 (given)
AOC = 2x + 2y = 2(5) + 2(10) = 20
AOD = 2x + 2y + 2z = 2(5) + 2(10) + 2(10) = 50

16. Similarly, for each valid combination, we can find the values of AOC and AOD.

Therefore, to find angle AOC and angle AOD, we need to solve the equation x + y + z = 25 and check all possible combinations of y and z.