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Section 10-2 Measuring Angles and ArcsMonday, May 14, 2012
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Essential Questions • How do you identify central angles, major arcs, minor arcs, and semicircles, and ﬁnd their measures? • How do you ﬁnd arc length?Monday, May 14, 2012
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Vocabulary 1. Central Angle: 2. Arc: 3. Minor Arc: 4. Major Arc:Monday, May 14, 2012
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Vocabulary 1. Central Angle: An angle inside a circle with the vertex at the center and each side is a radius 2. Arc: 3. Minor Arc: 4. Major Arc:Monday, May 14, 2012
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Vocabulary 1. Central Angle: An angle inside a circle with the vertex at the center and each side is a radius 2. Arc: A part of a exterior of the circle 3. Minor Arc: 4. Major Arc:Monday, May 14, 2012
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Vocabulary 1. Central Angle: An angle inside a circle with the vertex at the center and each side is a radius 2. Arc: A part of a exterior of the circle 3. Minor Arc: An arc that is less than half of a circle; Has same measure as the central angle that contains it 4. Major Arc:Monday, May 14, 2012
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Vocabulary 1. Central Angle: An angle inside a circle with the vertex at the center and each side is a radius 2. Arc: A part of a exterior of the circle 3. Minor Arc: An arc that is less than half of a circle; Has same measure as the central angle that contains it 4. Major Arc: An arc that is more than half of a circle; Find the measure by subtracting the measure of the minor arc with same length from 360°Monday, May 14, 2012
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Vocabulary 5. Semicircle: An arc that is half of a circle; the measure of a semicircle is 360° 6. Congruent Arcs: 7. Adjacent Arcs:Monday, May 14, 2012
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Vocabulary 5. Semicircle: An arc that is half of a circle; the measure of a semicircle is 360° 6. Congruent Arcs: Arcs that have the same measure 7. Adjacent Arcs:Monday, May 14, 2012
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Vocabulary 5. Semicircle: An arc that is half of a circle; the measure of a semicircle is 360° 6. Congruent Arcs: Arcs that have the same measure 7. Adjacent Arcs: Two arcs in a circle that have exactly one point in commonMonday, May 14, 2012
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Theorems and Postulates Theorem 10.1 - Congruent Arcs: Postulate 10.1 - Arc Addition Postulate: Arc Length:Monday, May 14, 2012
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Theorems and Postulates Theorem 10.1 - Congruent Arcs: In the same or congruent circles, two minor arcs are congruent IFF their central angles are congruent Postulate 10.1 - Arc Addition Postulate: Arc Length:Monday, May 14, 2012
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Theorems and Postulates Theorem 10.1 - Congruent Arcs: In the same or congruent circles, two minor arcs are congruent IFF their central angles are congruent Postulate 10.1 - Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs Arc Length:Monday, May 14, 2012
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Theorems and Postulates Theorem 10.1 - Congruent Arcs: In the same or congruent circles, two minor arcs are congruent IFF their central angles are congruent Postulate 10.1 - Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs D Arc Length: l = i2π r 360Monday, May 14, 2012
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Example 1 Find the value of x when m∠QTV = (20x)°, m∠QTR = 20°, m∠RTS = (8x − 4)°, m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°.Monday, May 14, 2012
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Example 1 Find the value of x when m∠QTV = (20x)°, m∠QTR = 20°, m∠RTS = (8x − 4)°, m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°. 20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360Monday, May 14, 2012
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Example 1 Find the value of x when m∠QTV = (20x)°, m∠QTR = 20°, m∠RTS = (8x − 4)°, m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°. 20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360 46x + 38 = 360Monday, May 14, 2012
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Example 1 Find the value of x when m∠QTV = (20x)°, m∠QTR = 20°, m∠RTS = (8x − 4)°, m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°. 20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360 46x + 38 = 360 46x = 322Monday, May 14, 2012
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Example 1 Find the value of x when m∠QTV = (20x)°, m∠QTR = 20°, m∠RTS = (8x − 4)°, m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°. 20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360 46x + 38 = 360 46x = 322 x=7Monday, May 14, 2012
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Example 2 WC is the radius of ⊙C. Identify each as a major arc, minor arc, or semicircle. Then ﬁnd each measure. a. XZY b. WZX c. XWMonday, May 14, 2012
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Example 2 WC is the radius of ⊙C. Identify each as a major arc, minor arc, or semicircle. Then ﬁnd each measure. a. XZY Semicircle, 180° b. WZX c. XWMonday, May 14, 2012
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Example 2 WC is the radius of ⊙C. Identify each as a major arc, minor arc, or semicircle. Then ﬁnd each measure. a. XZY Semicircle, 180° b. WZX Major arc, 270° c. XWMonday, May 14, 2012
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Example 2 WC is the radius of ⊙C. Identify each as a major arc, minor arc, or semicircle. Then ﬁnd each measure. a. XZY Semicircle, 180° b. WZX Major arc, 270° c. XW Minor arc, 90°Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% a. Find the measure of the arc of Comfort the section that represents the 21% Youth Hybrid comfort bicycles. 26% 9% Other 7% Mountain 37%Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% a. Find the measure of the arc of Comfort the section that represents the 21% Youth Hybrid comfort bicycles. 26% 9% Other 7% 360(.21) Mountain 37%Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% a. Find the measure of the arc of Comfort the section that represents the 21% Youth Hybrid comfort bicycles. 26% 9% Other 7% 360(.21 = 75.6° ) Mountain 37%Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% b. Find the measure of the arc Comfort representing the combination of 21% Youth Hybrid the mountain, youth, and comfort 26% 9% bicycles. Other 7% Mountain 37%Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% b. Find the measure of the arc Comfort representing the combination of 21% Youth Hybrid the mountain, youth, and comfort 26% 9% bicycles. Other 7% 360(.37 + .26 + .21) Mountain 37%Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% b. Find the measure of the arc Comfort representing the combination of 21% Youth Hybrid the mountain, youth, and comfort 26% 9% bicycles. Other 7% 360(.37 + .26 + .21 = 360(.84) ) Mountain 37%Monday, May 14, 2012
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Example 3 Refer to the table showing the percent of bicycles bought by type at a bike shop. Type Mountain Youth Comfort Hybrid Other Percent 37% 26% 21% 9% 7% b. Find the measure of the arc Comfort representing the combination of 21% Youth Hybrid the mountain, youth, and comfort 26% 9% bicycles. Other 7% 360(.37 + .26 + .21 = 360(.84) ) Mountain 37% = 302.4°Monday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL b. mHJ c. mLH d. mKJMonday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 b. mHJ c. mLH d. mKJMonday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 = 328° b. mHJ c. mLH d. mKJMonday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 = 328° b. mHJ = 180 − 32 c. mLH d. mKJMonday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 = 328° b. mHJ = 180 − 32 = 148° c. mLH d. mKJMonday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 = 328° b. mHJ = 180 − 32 = 148° c. mLH = 32° d. mKJMonday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 = 328° b. mHJ = 180 − 32 = 148° c. mLH = 32° d. mKJ = 90 + 32Monday, May 14, 2012
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Example 4 Find the measure of each arc. a. mKHL = 360 − 32 = 328° b. mHJ = 180 − 32 = 148° c. mLH = 32° d. mKJ = 90 + 32 = 122°Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. a.Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. a. D l= i2π r 360Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. a. D l= i2π r 360 40 = i2π (4.5) 360Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. a. D l= i2π r 360 40 = i2π (4.5) 360 ≈ 3.14 cmMonday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. b.Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. b. D l= i2π r 360Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. b. D l= i2π r 360 152 = i2π (6) 360Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. b. D l= i2π r 360 152 = i2π (6) 360 ≈ 15.92 cmMonday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. c.Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. c. D l= i2π r 360Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. c. D l= i2π r 360 140 = i2π (6) 360Monday, May 14, 2012
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Example 5 Find the length of DA , rounding to the nearest hundredth. c. D l= i2π r 360 140 = i2π (6) 360 ≈ 14.66 cmMonday, May 14, 2012
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Check Your Understanding p. 696 #1-11Monday, May 14, 2012
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Problem Set p. 696 #13-41 odd, 55, 73 "Our lives improve only when we take chances - and the ﬁrst and most difﬁcult risk we can take is to be honest with ourselves." - Walter AndersonMonday, May 14, 2012
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