Angles, Arcs, and Chords Advanced Geometry Circles Lesson 2.
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Transcript of Angles, Arcs, and Chords Advanced Geometry Circles Lesson 2.
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