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Geometry - Circles

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    Study Guide Geometry – Circles 1. Central Angles and Arcs Circles have several different types of angles. The most important one to start with is the central angle . A central angle is an angle whose vertex (corner point) is at the center of the circle. Its sides are radii of the circle. A full circle measures 360° . That means a central angle can “sweep” across part of the circle, and its measure tells us how much of the circle is covered. Figure 1 In the figure, ∠ AOB is a central angle because: • Point O is the center. • OA and OB are radii. 1.1 Understanding Arcs An arc is a part of the circle’s edge.

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    Study Guide It includes: • Two endpoints • All the points between them along the circle We use a curved line symbol over the letters to name an arc. There are three types of arcs : Semicircle A semicircle is half of a circle. • It measures 180° . • Its endpoints are the endpoints of a diameter . • It is named using three letters . The first and last letters are the endpoints of the diameter. The middle letter is any point on that arc. Figure 2 If AC is a diameter, then arc ABC is a semicircle. Minor Arc A minor arc is smaller than a semicircle. • Its measure is less than 180° . • It is named using two letters (just the endpoints).

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    Study Guide Figure 3 Major Arc A major arc is larger than a semicircle. • Its measure is more than 180° . • It is named using three letters . The first and last letters are the endpoints. The middle letter is a point somewhere along the arc. Figure 4 1.2 Measuring Arcs Arcs can be measured in two ways: 1. Degree measure 2. Arc length (units like inches or cm) Let’s look at each type.

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    Study Guide Degree Measure of a Semicircle A semicircle always measures: Its length is half of the circumference . Degree Measure of a Minor Arc The measure of a minor arc is the same as its central angle . Its length is a portion of the circumference and is always less than half . Degree Measure of a Major Arc To find a major arc: Its length is always more than half the circumference. Important Notation Example 1: Solve Circle O has diameter AB and OB = 6 inches. Find: • (a) the degree measure of arc AXB • (b) the length of arc AXB

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    Study Guide Figure 5 Since AXB is a semicircle: To find the length: Circumference formula: Here, radius = 6 Full circumference: Half of that: So the arc length is:

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    Study Guide 1.3 Arc Addition Postulate If point B is between A and C on a circle, then: This simply means: Small arcs add up to make a larger arc. Example 2: Find If: Using the Arc Addition Postulate: Figure 6

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    Study Guide Example 3: Solve Circle P has diameter QS. Find: Figure 7 Key ideas used: • Minor arc = central angle • Semicircle = 180° • Major arc = 360° − minor arc 1.4 Theorems About Arcs and Central Angles Theorem 1: If two central angles are equal, then their corresponding minor arcs are equal. Theorem 2: If two minor arcs are equal, then their corresponding central angles are equal.

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    Study Guide Example 4: Solve Circle O has diameters AC and BD. If ( m ∠ 1 = 40° ), find the remaining measures. Figure 8 Key results: • Minor arc equals its central angle. • Vertical angles are equal. • A semicircle measures 180°. • The angles of a triangle add to 180°. After solving: • Some arcs measure 40° • Some measure 140° • ∠ 3 and ∠ 4 both equal 20° Summary • A central angle has its vertex at the center. • Minor arc < 180° • Semicircle = 180°

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    Study Guide • Major arc > 180° • Minor arc measure = central angle • Major arc = 360° − minor arc • Arcs can be added together 2. Arcs and Inscribed Angles So far, we’ve talked about central angles — angles whose vertex is at the center of a circle. But central angles are not the only angles found in circles. In fact, many important angles are formed on the circle itself . 2.1 What Is an Inscribed Angle? An inscribed angle is: • An angle formed by two chords • With its vertex on the circle This is different from a central angle, whose vertex is at the center. 2.2 What Is an Intercepted Arc? Every inscribed angle “cuts off” a part of the circle. That part is called the intercepted arc . It includes: • The endpoints of the angle • The portion of the circle between those endpoints

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    Study Guide Figure 1 In this figure: • ∠ ABC is an inscribed angle. • Arc AC is its intercepted arc. 2.3 Angles That Are NOT Inscribed Angles Not every angle inside a circle is inscribed. For an angle to be inscribed: • Its vertex must lie on the circle . • Its sides must be chords . If the vertex is inside the circle but not on it, the angle is NOT inscribed. Figure 2

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