# Modern School Mathematics Geometry Chapter 10

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Circles

### Tangent

a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point

### Point of Tangency

the point where a circle and a tangent intersect

### Theorem 10-1

A line in the plane of a circle and containing an interior point of the circle intersects the circle in two points.

### Theorem 10-2

In a circle, tangent perpendicular to the radius

### Theorem 10-3

In a circle, tangent perpendicular to radius

### Common tangent between two circles

is tangent to both

### Common internal tangent

crosses through space in the middle

### Common external tangent

does not cross through space in the middle

### Statements that can be used

Radii of a circle are congruent
In a circle, radii congruent to radii of congruent circles are congruent

### Circle

The set of all points equally distant from a given point

### Central Angle

An angle in a circle with the vertex on the center

### Minor Arc

An arc of a circle whose measure is less than 180 degrees (less then a semicircle)

### Major Arc

part of a circle that measures from 180 to 360 (greater than a semicircle)

### Semicircle

Half a circle, the arc created by the endpoints of the diameter

### Measure of minor arc

the measure of its central angle

### Measure of major arc

the difference between 360 and the measure of its associated minor arc

180 degrees

### Congruent Arcs

In congruent circles, have the same measure

If the intersection of arcs DE and EF of a circle is the single point E, then mDE+mEF=mDEF

### Theorem 10-4

In same or congruent circles, congruent central angles yield congruent arcs

### Theorem 10-5

In same or congruent circles, congruent arcs yield congruent central angles

### Chords

a segment that both endpoints on the circle

### Arc of a chord

The part of the circle cut off by those endpoints

### Theorem 10-6

In same or congruent circles, congruent chords yield congruent arcs

### Theorem 10-7

In same or congruent circles, congruent arcs yield congruent chords

### Theorem 10-8

Diameters perpendicular to a chord yield bisected chord and arcs

### Theorem 10-9

In same or congruent circles, chords equidistant from center arcs are congruent

### Theorem 10-10

In same or congruent circles, congruent chords are equidistant from the center

### Inscribed angle

An angle in the circle whose vertex is on the circle, and both sides (rays) intersect the circle

### An angle intercepts and arc if

1. The vertex is on the circle
2. Both rays intersect the circle

### Theorem 10-11

m(inscribed ange)= 1/2 intercepted arc

### Corollary 1

An angle inscribed in a semicircle is a right angle

### Corollary 2

If a quadrilateral is inscribed in a circle, that yields supplementary opposite angles.

### Corollary 3

In a circle, congruent ars yield congruent inscribed angles

### Secant

a line that intersects a circle at two points. A line that contains a chord.

### Theorem 10-13

In a circle, m(interior angle) = the average of the intercepted arcs or:
1/2(∑ or intercepted arcs)

### Theorem 10-14

In a circle, m(exterior ∠) = ½ the difference of the intercepted arcs, or:
In a circle, m(exterior ∠) = ½(arc₂-arc₁)
Secant/Secant

### Theorem 10-15

tan/secant= ½(arc₂-arc₁)

### Theorem 10-16

If the chords bisect inside the circle, product of 2 part of one chord=the product of 2 parts of other chord (ab=xy)

### Theorem 10-17

If secants intersect out side the circle:
(whole part₁)(outside₁)=(whole part₂)(outside₂)

### Theorem 10-18

tan²= (whole part)(outside)

### Important!

From a point outside a circle, both tangents are congruent.

Example: