Transcript 9.2: Angles and Arcs

4.1a: Central/Inscribed Angles
in
Circles
CCSS:
G-C.2
Identify and describe relationships among inscribed angles, radii, and chords. Include
the relationship between central, inscribed, and circumscribed angles; inscribed
angles on a diameter are right angles; the radius of a circle is perpendicular to the
tangent where the radius intersects the circle.
G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
GSE’s
M(G&M)–10–2 Makes and defends conjectures, constructs geometric
arguments, uses geometric properties, or uses theorems to solve problems
involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine,
tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean
Theorem, Triangle Inequality Theorem).
What is a circle?
the set of all points in a plane that
are equidistant from a given point
Central Angle: an angle whose vertex is at the center of the circle
A
Circle B
ABC
Has a vertex at the center
B
C
Sum of Central Angles: The sum of all central angles in a circle
Is 360 degrees.
A
Find m ADC
D
B
80
C
Little m indicates degree measure of the arc
AC is a minor arc. Minor arcs are less than 180 degrees. They use the
the two endpoints.
ADC is a major arc. Major arc are greater than 180 degrees. They use
three letters, the endpoints and a point in-between them.
Major Concept: Degree measures of arcs are the same as its central angles
What is the mFY?
What is the mFRY?
NECAP type question
Circle P has a diameter added to its figure every step
so all central angles are congruent.
What is the sum of the measures of 3 central angles after
the 5th step? Explain in words how you know.
Step 2
Step 1
Step 3
In Circle P
m1  140 with diameter AC. Find each measure.
1) m2

2) m BC

3) m AB

4) m ABC
5) mPAB
In circle F, m  EFD = 4x+6, m  DFB = 2x + 20. Find mAB
NECAP Released Item 2009
Inscribed Angle: An angle with a vertex ON the circle and made up of 2 chords
ABC Is the inscribed angle
Intercepted Arc: The arc formed by connecting the two endpoints
of the inscribed angle
Major Concept:
Inscribed angles degree measures are half the degree measure of
their intercepted arc
Ex
What is mACB
What is the mBG
What is the mGCB?
Major Concept:
If 2 different inscribed angles intercept the same arc, then
the angles are congruent
find mACB
and mAGB
Important Fact: If a quadrilateral is inscribed in a circle, then the opposite angles
are SUPPLEMENTARY
What angles are supplementary
Example:
Circle C, mS  28 and mR  110
Find mQ and mT
Find the degree measure
of all angles and arcs
Concentric Circles- circles with the same center, but different Radii
What is an example you can think of outside of geometry?