Q1. Complete the statements below by filling in the missing word(s) so that the statements are CORRECT:

1. The angle subtended by a chord at the centre of a circle is ...

2. The angle between the tangent and a chord is ...

3. The opposite angles of a cyclic quadrilateral are ...

Background

Topic: Circle Geometry (Euclidean Geometry)

This question tests your understanding of fundamental properties and theorems related to circles, including angles subtended by chords, properties of tangents, and cyclic quadrilaterals. These are common topics in trigonometry and geometry, especially in problems involving circles.

Key Terms and Theorems:

• Chord: A straight line segment whose endpoints both lie on the circle.

• Angle subtended by a chord at the centre: The angle formed at the centre of the circle by lines drawn from the centre to the endpoints of the chord.

• Tangent: A straight line that touches the circle at exactly one point.

• Cyclic Quadrilateral: A quadrilateral with all its vertices on the circumference of a circle.

Key Theorems:

• Angle at the Centre Theorem: The angle subtended by a chord at the centre of a circle is twice the angle subtended at the circumference on the same side of the chord.

• Tangent-Chord Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

• Cyclic Quadrilateral Theorem: The sum of the measures of the opposite angles of a cyclic quadrilateral is 180 degrees.

Step-by-Step Guidance

1. Recall the theorem about the angle subtended by a chord at the centre of a circle. Think about how this angle compares to the angle subtended by the same chord at the circumference.

2. For the tangent and chord, remember the relationship between the angle formed by a tangent and a chord at the point of contact and the angle in the alternate segment of the circle.

3. For cyclic quadrilaterals, use the property that relates the measures of opposite angles. Consider what their sum should be for any quadrilateral inscribed in a circle.

Try solving on your own before revealing the answer!