Angles, Arcs, and Chords

Advanced GeometryCircles

Lesson 2

In a circle or in congruent circles, two minor arcs are congruent if their corresponding chords are congruent.

Inscribed & Circumscribed

Quadrilateral ABCD is

inscribed in X.

X is circumscribed

about quadrilateral

ABCD.

inside surrounding

ALL vertices of the polygon

must lie on the circle.

Example: A circle is circumscribed about a regular pentagon. What is the measure of the arc between each pair of consecutive vertices?

In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the

chord and its arc.

is perpendicular to chord

Example: Circle R has a radius of 16 centimeters. Radius RU

TV

TV UV

, which is 22 cm long.

If m = 110, find m .

Find RS.

= 53, find m .

is perpendicular to chord

Example: Circle W has a radius of 10 centimeters. Radius WL

HKHL MK

, which is 16 cm long.

If m

Find JL.

EF GH

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

the center.

and are equidistant from the center. P is 15 and EF = 24, find PR and RH.

Example: Chords

If the radius of

Inscribed Angles & Intercepted Arcs

is intercepted by RT RST

is an inscribed angleRST

If an angle is inscribed in a circle, then the measure of the angle equals one-half the

measure of its intercepted arc.

1

2m RST mRT

= 100.Find m 1, m 2, m 3, m 4, and m 5.

AD BC

Example: In O, m = 140, m

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

Example: Find m 2 if ∠ m 2 = 5x – 6 and ∠ m 1 = 3x + 18. ∠

Example: Triangles TVU and TSU are inscribed in P with .

VU SU

If the inscribed angle intercepts a semicircle, then the angle is a right angle.

Find the measure of each numbered angle if m 2 = x + 9∠and m 4 = 2x + 6.∠

S and m

Example: Quadrilateral QRST is inscribed in

If a quadrilateral is inscribed in a circle, then itsopposite angles are supplementary.

M. If m Q = 87R = 102, T.find mand m

Example: Points M and N are on a circle so that m

Suppose point L is randomly located on the same circle so that it does not coincide with M or N. What is the probability that m

MN

∠MLN = 40?

Probability

= 80.

size of the eventProbability of an event =

size of the sample space