4.1a: central/inscribed angles in circles m(g&m)–10–2 makes and defends conjectures,...

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4.1a: Central/Inscribed Angles in Circles 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). GSE’s G -C.2 Identify and describe relationshipsam ong 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. CCSS: G -C.3 Constructthe inscribed and circum scribed circlesofa triangle , and prove propertiesof anglesfora quadrilateralinscribed in a circle.

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Page 1: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

4.1a: Central/Inscribed Angles in Circles

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).

GSE’s

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.

CCSS:

G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Page 2: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

What is a circle?

the set of all points in a plane that are equidistant from a given point

Page 3: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Central Angle: an angle whose vertex is at the center of the circle

A

B

C

ABCCircle B

Has a vertex at the center

Sum of Central Angles: The sum of all central angles in a circle Is 360 degrees.

A

B

C

80

Find m ADC

D

Little m indicates degree measure of the arc

Page 4: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

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.

Page 5: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Major Concept: Degree measures of arcs are the same as its central angles

What is the mFY?

What is the mFRY?

Page 6: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

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 1Step 2

Step 3

NECAP type question

Page 7: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

In Circle P

PAB

m

m )5

ABC m )4

AB m )3

BC m 2)

2m 1)

measure.each Find .ACdiameter with 1401

Page 8: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

In circle F, m EFD = 4x+6, m DFB = 2x + 20. Find mAB

Page 9: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

NECAP Released Item 2009

Page 10: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

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

Page 11: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Major Concept:

Inscribed angles degree measures are half the degree measure of their intercepted arc

Ex

What is ACBm

Page 12: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

What is the mBG

What is the mGCB?

Page 13: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Major Concept: If 2 different inscribed angles intercept the same arc, thenthe angles are congruent

AGBm and

ACBm

find

Page 14: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Important Fact: If a quadrilateral is inscribed in a circle, then the opposite angles are SUPPLEMENTARY

What angles are supplementary

Page 15: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Example:Circle C,

Tm and Qm Find

110Rm and 28

Sm

Page 16: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Find the degree measure of all angles and arcs

Page 17: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Page 18: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

Concentric Circles- circles with the same center, but different Radii

What is an example you can think of outside of geometry?

Page 19: 4.1a: Central/Inscribed Angles in Circles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses

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