4.1a: Central/Inscribed Angles in Circles

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Presentation transcript:

4.1a: Central/Inscribed Angles in Circles CCSS: 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 Circle B 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 80 B D Little m indicates degree measure of the arc C

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

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 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

What is the mBG What is the mGCB?

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

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

Example: Circle C,

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?