In AB and CD are two equal chords of a circle with center O. OP and OQ are perpendiculars on chords AB and CD respectively. If `anglePOQ=150^(@)` then `angleAPQ` is equal to

Options

A. `30^(@)`
B. `75^(@)`
C. `15^(@)`
D. `60^(@)`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript and break it down into clear steps.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have two equal chords AB and CD in a circle with center O.
- OP and OQ are perpendiculars drawn from the center O to the chords AB and CD, respectively.
- We are given that \( \angle POQ = 150^\circ \).

2. **Identifying Equal Distances**:
- Since AB and CD are equal chords, the perpendicular distances from the center O to these chords are also equal. Therefore, \( OP = OQ \).

3. **Setting Up Angles**:
- Let \( \angle BOP = \angle AOP = \theta \) (angles formed by the radius and the perpendicular).
- Since OP and OQ are equal, the triangles formed (triangle OAP and triangle OBP) are isosceles triangles. Thus, \( \angle APQ = \angle BQP \).

4. **Using the Triangle Angle Sum Property**:
- In triangle POQ, the sum of the angles is \( 180^\circ \):
\[ \angle POQ + \angle OPQ + \angle OQP = 180^\circ \]
- Since \( \angle POQ = 150^\circ \), we have:
\[ 150^\circ + \angle OPQ + \angle OQP = 180^\circ \]
- This simplifies to:
\[ \angle OPQ + \angle OQP = 30^\circ \]

5. **Finding Each Angle**:
- Since \( \angle OPQ = \angle OQP \) (because they are angles opposite to equal sides in triangle OPQ), we can let:
\[ \angle OPQ = \angle OQP = x \]
- Thus, we have:
\[ 2x = 30^\circ \implies x = 15^\circ \]
- Therefore, \( \angle OPQ = 15^\circ \) and \( \angle OQP = 15^\circ \).

6. **Finding Angle APQ**:
- Now consider the straight line AB:
\[ \angle BPO + \angle APQ + \angle OPB = 180^\circ \]
- Since \( \angle BPO = 90^\circ \) (because OP is perpendicular to AB), we have:
\[ 90^\circ + \angle APQ + 15^\circ = 180^\circ \]
- This simplifies to:
\[ \angle APQ = 180^\circ - 105^\circ = 75^\circ \]

### Final Answer:
The measure of \( \angle APQ \) is \( 75^\circ \).

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