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Notes – Circle Basics Name: M Standard: Period:

RECALL

EXAMPLES Give an example of each of the following:

1. Name the circle 2. Radius

3. Chord 4. Diameter

5. Secant 6. Tangent (line)

7. Point of tangency 8. Tangent (segment)

DEFINTION

A ________________ _____________ of a circle is an angle whose vertex is the center of the circle.

Minor Arcs Major Arcs < 180° > 180° Named by endpoints of the arc

Named by endpoints and one other point

A _________________________ is an arc with endpoints that are the endpoints of a diameter.

EXAMPLES Determine if each is a minor arc, major arc, or semicircle. Find the measure of each.

9. 𝑅�̂� 10. 𝑅𝑇�̂� 11. 𝑅𝑆�̂�

Geometry msheizer.weebly.com Similarity 12. Name a major arc.

13. Name a minor arc.

POSTULATE – Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of

the measures of the two arcs.

𝑚𝐴𝐵�̂� = 𝑚𝐴�̂� + 𝑚𝐵�̂�

EXAMPLES Identify the given arc as a major arc, minor arc, or semicircle, then find the measure of each arc.

14. 𝑇�̂� 15. 𝑄𝑅�̂�

16. 𝑇𝑄�̂� 17. 𝑄�̂�

18. 𝑇�̂� 19. 𝑅𝑆�̂�

CONGRUENT CIRCLES CONGRUENT ARCS Two circles are congruent if they have the same radius Two arcs are congruent if they have the same measure

and they are arcs of the same circle or of congruent circles.

EXAMPLES Are the arcs congruent? Explain.

20.

21.

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WKS – Circle Basics Name: M Standard: Period:

SHOW ALL WORK!

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Notes – Inscribed Angles & Polygons Name: Th Standard: Period:

DEFINITION

An ________________________ angle is an angle whose vertex is on a

circle and whose sides contain chords of the circle.

The arc that lies in the interior of an inscribed angle and has endpoints

on the angle is called the _____________________ arc of the angle.

THEOREM

The measure of an inscribed angle is one half the measure of its intercepted arc.

𝑚∠𝐴𝐷𝐵 =1

2𝑚𝐴�̂� OR 𝑚𝐴�̂� = 2 ∙ 𝑚∠𝐴𝐷𝐵

EXAMPLES Find the indicated measure.

1. 𝑚∠𝑇 2. 𝑚𝑅�̂�

3. 𝑚𝑄�̂� 4. 𝑚∠𝑆𝑇𝑅

THEOREM

If two inscribed angles of a circle intercept the same arc, then the

angles are congruent.

EXAMPLES Find the indicated measure.

5. 𝑚∠𝐻𝐺𝐹

6. 𝑚𝑇�̂�

7. 𝑚∠𝑍𝑋𝑊

Geometry msheizer.weebly.com Similarity DEFINITION

A polygon is an _______________________ polygon if all

of its vertices lie on a circle.

The circle is said to _________________________ the

polygon.

THEOREM

A right triangle is inscribed in a circle if and only if the hypotenuse

is a diameter of the circle.

THEOREM

A quadrilateral can be inscribed in a circle if and only if its opposite

angles are supplementary.

EXAMPLES Find the value of each variable

1.

2.

3.

4.

Geometry msheizer.weebly.com Similarity

WKS – Inscribed Angles & Polygons Name: Th Standard: Period:

Find the measure of each variable, given arc, or given angle. SHOW ALL WORK!

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11.

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Geometry msheizer.weebly.com Similarity

Notes – Properties of Chords Name: M Standard: Period:

What is a chord?

NOTE

Any chord divides a circle into two arcs, the major arc and the minor arc.

A diameter divides a circle into two semicircles.

THEOREM

In the same circle, or in congruent circles, two minor arcs are congruent

if and only if their corresponding chords are congruent.

𝐴�̂� ≅ 𝐶�̂� 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

EXAMPLES

1. Find 𝑚𝐹�̂�. 2. If 𝑚𝐴�̂� = 110°, find 𝑚𝐵�̂�.

3. If 𝑚𝐴�̂� = 150°, find 𝑚𝐴�̂�.

Find the measure of each arc of a circle circumscribed around the following regular polygons. 4. Square 5. Hexagon 6. Nongon

Geometry msheizer.weebly.com Similarity THEOREMS – Diameters, Chords, and Perpendicular Bisectors

If one chord is a perpendicular bisector of another, then

the first is a diameter.

If a diameter of a circle is perpendicular to a chord, then

the diameter bisects the chord and its arc.

EXAMPLES CF = 7. Find the measure of each arc or segment.

7. EF 8. EC 9. 𝐶�̂�

10. 𝐷�̂� 11. 𝐶�̂�

THEOREM

In the same circle, or in congruent circles, two chords are

congruent if and only if they are equidistant from the center.

(perpendicular to radius)

𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐸𝐹 = 𝐸𝐺

EXAMPLES Find the indicated values

12. CU 13. QU

14. The radius of ⊙C

F

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WKS – Chords Name: M Standard: Period:

Find the indicated measure.

1. 𝑚𝐴�̂�

2. 𝑚𝐴�̂�

3. 𝐸𝐺

Solve for x. 4.

5.

6.

7.

8. *Note* 3x + 2 = ½ the chord AD

9.

Geometry msheizer.weebly.com Similarity What can you conclude about the diagram shown?

10.

11.

12.

13. In the diagram of ⊙R, which congruence relationship is not necessarily true?

Find the measure of each arc of a circle circumscribed about the regular polygon. 14. Triangle 15. Pentagon 16. Octagon

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Notes – Tangents Name: W Standard: Period:

RECALL

A tangent is any line or segment that touches the edge of a circle in exactly one spot.

THEOREM

In a plane, a line is tangent to a circle if and only if the line is

perpendicular to a radius of the circle at its endpoint on the circle.

Line m is tangent to ⊙ 𝑄 if and only if 𝑚 ⊥ 𝑄𝑃̅̅ ̅̅

Examples

1. Given that 𝑃𝑇̅̅̅̅ is a radius, determine if 𝑆𝑇̅̅̅̅ is tangent to the circle.

2. Find the value of r so that 𝐴𝐵̅̅ ̅̅ is tangent to the circle and 𝐶𝐵̅̅ ̅̅ is a radius.

Geometry msheizer.weebly.com Similarity THEOREM

Tangent segments from a common external point are congruent. (Ice Cream Cone Theorem)

If 𝑆𝑇̅̅̅̅ and 𝑆𝑅̅̅̅̅ are tangents, then 𝑆𝑇̅̅̅̅ ≅ 𝑆𝑅̅̅̅̅ .

EXAMPLES Find the value of x.

3.

4.

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WKS – Tangents Name: W Standard: Period:

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Notes – More Angles Name: Th Standard: Period:

THEOREM

If a tangent and a chord intersect at a point on a circle, then the measure of

each angle is one half the measure of intercepted arc.

Examples

1. 𝑚∠1

2. 𝑚𝑅𝑆�̂�

3. 𝑚𝑋�̂�

THEOREM

If an angle is on the outside of a circle, then the measure of the angle formed is one half the difference of the

measures of the intercepted arcs.

𝑚∠1 =1

2(𝑚𝐵�̂� − 𝑚𝐴�̂�) 𝑚∠2 =

1

2(𝑚𝑃𝑄�̂� − 𝑚𝑃�̂�) 𝑚∠3 =

1

2(𝑚𝑋�̂� − 𝑚𝑊�̂�)

Examples

4.

5.

6.

Geometry msheizer.weebly.com Similarity 7.

8.

Geometry msheizer.weebly.com Similarity

WKS – Angle Practice Name: Th Standard: Period:

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Geometry msheizer.weebly.com Similarity

Notes – Equations of Circles Name: F Standard: Period:

EQUATION

The equation of a circle with center (h, k) and a radius of r units is:

(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2

Examples Write an equation for the circle with the given center and radius. Then graph the circle.

1. Center at (0, 0) and r = 1 2. Center at (2, -4) and r = 1

Find the circle’s center and radius, then graph the equation.

3. (𝑥 + 1)2 + (𝑦 − 2)2 = 9 4. (𝑥 − 2)2 + (𝑦 − 1)2 = 4 5. 𝑥2 + 𝑦2 = 16

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WKS – Equations of Circles Name: F Standard: Period: