If AB is a chord of a circle, P and Q are two points on the circle different from A and B, then:

Options

A. the angle subtended by AB at P and Q are either equal or supplementary.
B. the sum of the angles subtended by AB at P and Q alwasys equal two right angles.
C. the angles subtended at P and Q by AB are always equal.
D. the sum of the angles subtended at P and Q is equal to four right angles.

Answer

To solve the problem, we need to analyze the properties of angles subtended by a chord in a circle. Here’s a step-by-step breakdown of the solution:

### Step 1: Draw the Circle and Chord
Draw a circle and label it. Identify a chord within the circle and label the endpoints of the chord as points A and B.

**Hint:** Remember that a chord is a straight line connecting two points on the circumference of a circle.

### Step 2: Identify Points on the Circle
Select two additional points on the circumference of the circle, labeling them as points P and Q. Ensure that P and Q are distinct from points A and B.

**Hint:** The points P and Q should be anywhere on the circumference, but not on the chord itself.

### Step 3: Draw Angles
Draw line segments from point A to point P and from point B to point P. This creates angle APB. Similarly, draw line segments from point A to point Q and from point B to point Q, creating angle AQB.

**Hint:** The angles APB and AQB are formed by the chord AB and the lines connecting A and B to points P and Q.

### Step 4: Analyze the Angles
According to the properties of circles, the angles subtended by the same chord (AB) at any point on the circumference (P and Q) are equal. Therefore, angle APB is equal to angle AQB.

**Hint:** Recall the theorem that states the angle subtended by a chord at any point on the circumference is the same.

### Step 5: Conclusion
Since the angles APB and AQB are equal, we conclude that the correct statement from the options provided is that the angles subtended at points P and Q by chord AB are always equal.

**Final Answer:** The correct option is 3: "The angle subtended at P and Q by AB are always equal."