Question

1. which statement(s) are true about inscribed angles?

statement 1: the vertex of an inscribed angle lies on the circle.
statement 2: the sides of an inscribed angle are tangent lines to the circle.
statement 3: the sides of an inscribed angle are chords of a circle.
statement 4: the vertex of an inscribed angle lies outside a circle.
2 & 4 only
1 only
1 & 3 only
4 only

Explanation:

To determine the true statements about inscribed angles, we recall the definition of an inscribed angle: an angle whose vertex lies on the circle and whose sides are chords of the circle (since chords connect two points on the circle, and the sides of the inscribed angle must connect the vertex (on the circle) to other points on the circle).

• Statement 1: The vertex of an inscribed angle lies on the circle. This matches the definition, so it is true.
• Statement 2: The sides of an inscribed angle are tangent lines to the circle. Tangent lines touch the circle at only one point, but the sides of an inscribed angle must connect two points on the circle (as chords), so this is false.
• Statement 3: The sides of an inscribed angle are chords of a circle. Since chords are segments with both endpoints on the circle, and the vertex of the inscribed angle is on the circle, the sides (connecting the vertex to other points on the circle) are chords. This is true.
• Statement 4: The vertex of an inscribed angle lies outside a circle. By definition, the vertex lies on the circle, so this is false.

Thus, Statements 1 and 3 are true.

Answer:

1 & 3 only