Given:
- PQ is tangent to the circle with center O at the point B.
- Angle AOB = 100°.

To find:
The measure of angle ABP.

Solution:

Step 1: Understanding the problem

- We have a circle with center O and a tangent PQ at point B.
- Angle AOB is given as 100°.

Step 2: Identifying the relevant angles

- Let's label the angles in the diagram to better understand the problem.

- Angle AOB = 100° (given)
- Angle ABP = ?
- Angle BPO = ?
- Angle OPB = ?

Step 3: Understanding the properties of a tangent

- A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
- Therefore, angle BPO is a right angle (90°).

Step 4: Using the properties of a tangent

- Since angle BPO is a right angle, triangle BPO is a right triangle.
- We can use the properties of a right triangle to find angle OPB.

Step 5: Finding angle OPB

- In a right triangle, the sum of the angles is 180°.
- We know that angle BPO = 90°.
- Let's assume angle OPB = x.

- Therefore, angle OPB + angle BPO + angle PBO = 180°.
- x + 90° + angle PBO = 180°.
- angle PBO = 180° - x - 90°.
- angle PBO = 90° - x.

Step 6: Using the properties of a tangent (continued)

- We know that angle PBO is equal to angle ABP. (Alternate angles)
- Therefore, angle ABP = 90° - x.

Step 7: Finding angle ABP

- We know that angle AOB = 100°. (Given)
- Angle AOB is the sum of angles ABP and OPB.

- Therefore, angle ABP + angle OPB = 100°.
- 90° - x + x = 100°.
- 90° = 100°.

Step 8: Conclusion

- Since 90° does not equal 100°, we made an error in our calculations.
- Double-check the calculations and try again to find the correct value for angle ABP.

Step 9: Retry the calculations

- Let's assume angle OPB = y. (Retry the calculations from Step 5)
- angle BOA = angle BOP + angle POA = 90° + 100° = 190°. (Sum of angles in a triangle)
- angle ABP = angle BOA - angle BOP = 190° - 90° = 100°.

Step 10: Conclusion

- After retrying the calculations, we find that angle ABP = 100°.
- Therefore, the measure of angle ABP is 100°.

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