Circles

Question 1

tangent

Answer 1

line that intersects circle at one point only, perpendicular to radius from that point of intersection

Question 2

secant

Answer 2

line that intersects circle at two points

Question 3

2 types of arcs

Answer 3

minor: all points within segment/angle
major: all points in exterior of segment/angle

Question 4

sector

Answer 4

formed by two radii and segment made by arc

Question 5

segment

Answer 5

area inside segment made by arc and curved part of arc

Question 6

major segment

Answer 6

everything outside of a segment

Question 7

Arc Addition Postulate

Answer 7

If two arcs share an endpoint and no other points in common, then the two arcs and their measurements add up to equal the larger arc and its measurement.

Question 8

Central Angle/Arc Postulate

Answer 8

If two central angles are equal, then their corresponding arcs are equal, and vice versa.

Question 9

Arc/Chord Theorem

Answer 9

If two arcs are equal, their chords are equal, and vice versa.

Question 10

Center/Perpendicular Theorem

Answer 10

A line drawn through the center of a circle perpendicular to a chord meets at its midpoint and bisects the arc determined by the chord.

Question 11

2 Chord Theorems

Answer 11

In two congruent circles or in the same circle…

1. If two chords are equal, the chords are the same distance from the center.
2. if two chords are equally distant from the center, they are congruent.

Question 12

External Point/Tangent Theorem

Answer 12

The tangents drawn to a circle from an external point are equal, and they form equal angles with the line joining the point to the center of the circle.

Question 13

Tangent Circle Theorem

Answer 13

If two circles are tangent, their point of contact is on the line connecting their centers.

Question 14

Center/Midpoint Theorem

Answer 14

The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.

Question 15

Tangent and Radius Relationship

Answer 15

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

Question 16

Line Joining Circle Centers Theorem

Answer 16

If two circles intersect, the line joining their centers is the perpendicular bisector of the common chord.

Question 17

[photo] classification and measurement of ∠BAC

Answer 17

central angle; equal to intercepted arc (∠BAC = arc BC)

Question 18

[photo] classification and measurement of ∠BDC

Answer 18

inscribed angle; half of intercepted arc (∠BDC = 1/2 arc BC)

Question 19

[photo] classification and measurement of ∠BCD

Answer 19

chord and tangent angle; half of intercepted arc (∠BCD = 1/2 arc BC)

Question 20

[photo] classification and measurement of ∠BEF (for example)

Answer 20

two chord angle; half the sum of intercepted arcs (∠BEF = 1/2 (arc BF + arc CD))

Question 21

[photo] classification and measurement of ∠BDF

Answer 21

two secant angle; half the difference of intercepted arcs (∠BDF = 1/2 (arc BF - CE)

Question 22

[photo] classification and measurement of ∠BDF

Answer 22

tangent and secant angle; half the difference of intercepted arcs (∠BDF = 1/2 (arc BE- arc CE))

Question 23

[photo] classification and measurement of ∠BDF

Answer 23

two tangent angle; half the difference of intercepted arcs (∠BDF = 1/2 (arc CQE - arc CE))

Question 24

Inscribed Angle Theorem Corollaries (4)

Answer 24

1. Inscribed angles that intercept the same arc or equal arcs are equal.
2. Angles inscribed in a semicircle are right angles.
3. The opposite angles of a quadrilateral inscribed in a circle are supplementary.
4. If a side of an inscribed quadrilateral is extended, the exterior angle formed is equal to the opposite interior angle.

Question 25

Parallel Line Theorem

Answer 25

Parallel lines intercept congruent arcs on a circle.

Question 26

Segment/Chord Theorem

Answer 26

If two chords of a circle meet at a point inside the circle, the product of the segments of one chord is equal to the product of the segments of the other.

Question 27

Segment/Secant Theorem

Answer 27

If two secants of a circle meet at a point outside the circle, the product of one secant and its external segment is equal to the product of the other secant and its external segment.

Question 28

Segment/Tangent/Secant Theorem

Answer 28

If a tangent and a secant are drawn to a circle from an external point, the square of the tangent is equal to the product of the secant and its external segment.

Question 29

Cyclic Quadrilateral Theorem

Answer 29

If a quadrilateral is cyclic, then the opposite angles of the quadrilateral are supplementary. (and vice versa)