# Chapter 10 Theorems

## 17 terms

### Theorem 10.1

If a line is a tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency

### Theorem 10.2

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle

### Theorem 10.3

If two segments from the same exterior point are tangent to a circle, then they are congruent

### Theorem 10.4

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent

### Theorem 10.5

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

### Theorem 10.6

If one chord is a perpendicular bisector of another chord, then the first chord is a diameter

### Theorem 10.7

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center

### Theorem 10.8

If an angle is inscribed in a circle, then its measure half the measure of the intercepted arc

### Theorem 10.9

If two inscribed angles of a circle intercept the same arc, then the angles are congruent

### Theorem 10.10

An angle inscribed in a semicircle is a right angle

### Theorem 10.11

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary

### Theorem 10.12

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

### Theorem 10.13

If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

### Theorem 10.14

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs

### Theorem 10.15

If two chords intersect in the interior of the circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord

### Theorem 10.16

If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment

### Theorem 10.17

If a secant segment and a tangent segment share an endpoint outside of a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.