1 / 11

Warm-up

Warm-up. J. Name a segment congruent to JM. ML 2) Name an arc congruent to KL. Arc JK 3) Name an arc congruent to JN. Arc LN 4) Name a segment congruent to JB. KB, BN, or BP 5) Name the midpoint of JL. M. K. M. B. L. N. P. Chapter 9 Section 4. Inscribed Angles. Vocabulary.

duyen
Download Presentation

Warm-up

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Warm-up J Name a segment congruent to JM. ML 2) Name an arc congruent to KL. Arc JK 3) Name an arc congruent to JN. Arc LN 4) Name a segment congruent to JB. KB, BN, or BP 5) Name the midpoint of JL. M K M B L N P

  2. Chapter 9Section 4 Inscribed Angles

  3. Vocabulary Inscribed Angle-an angle whose vertex is on the circle and whose sides each contain chords of the circle. Intercepted Arc- An angle intercepts an arc if and only if each of the following conditions holds. 1) The endpoints of the arc lie on the angle. 2) All points of the arc, except the endpoints, are in the interior of the circle. 3) Each side of the angle contains an endpoint of the arc.

  4. Vocabulary cont. Theorem 9-4-If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. Theorem 9-5-If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent. Theorem 9-6-If an inscribed angle of a circle intercepts a semicircle, then the mangle is a right angle. Theorem 9-7-If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

  5. Ex 1: In the circle at the right, the measure of arc ST is 68. Find the measures of angle 1 and angle 2. Since arc ST is intercepted by both angle 1 and angle 2, and they are inscribed angles, the measure of angle 1 and angle 2 is ½ the measure of arc ST. <1 = ½ arc ST <1 = ½ (68) <1 = 34 = <2 R 1 S Q T 2 P

  6. Ex 2: In the circle Q, AC is a diameter, the measure of arc CD = 68, and the measure of arc BE = 96. Find m<ABC, m<BDE, m<CED, and the measure of arc AD. m<ABC Since <ABC intercepts a semicircle, <ABC is a right angle. m<ABC = 90. m<BDE Since <BDE is an inscribed angle m<BDE = ½ arc BE m<BDE = ½ (96) m<BDE = 48 m<CED Since <CED is an inscribed angle, m<CED = ½ arc CD m<CED = ½ (68) m<CED = 34 C B . Q A E D The measure of arc AD arc AC = arc AD + arc DC 180 = arc AD + 68 112 = arc AD

  7. Ex 3: In circle X, m<4 = 7x + 3, m<5 = 7x + 3 and m<1 = 5x. Find m<1, m<2, m<4, and m<5. Find x Since <6 intercepts a semicircle, m<6 = 90. Also the angles inside a triangle add up to 180. 180 = m<4 + m<5 + m<6 180 = 7x + 3 + 7x + 3 + 90 180 = 14x + 96 84 = 14x 6 = x m<1 m<1 = 5x m<1 = 5(6) m<1 = 30 6 . X 4 5 1 2 3 m<2 The angles inside a triangle add up to 180. 180 = m<1 + m<2 + m<3 180 = 30+ m<2 + 90 180 = m<2 + 120 60 = m<2 m<4 and m<5 m<4 = 7x + 3 m<4 = 7(6) + 3 m<4 = 42 + 3 m<4 = 45 = m<5

  8. Ex 4: In circle X, m<1 = 6x + 11, m<2 = 9x + 19, m<3 = 4y – 25, m<4 = 3y – 9, and arc PQ is congruent to arc RS. Find m<1, m<2, m<3, and m<4. Q Find x Since <PTS intercepts a semicircle, m<PTS = 90. Also the angles inside a triangle add up to 180. 180 = <PTS + m<1 + m<2 180 = 90 + 6x + 11 + 9x + 19 180 = 15x + 120 60 = 15x 4 = x m<1 m<1 = 6x + 11 m<1 = 6(4) + 11 m<1 = 24 + 11 m<1 = 35 R . X 4 3 S P 1 2 T m<2 m<2 = 9x + 19 m<2 = 9(4) + 19 m<2 = 36 + 19 m<2 = 55

  9. Ex 4 continued: In circle X, m<1 = 6x + 11, m<2 = 9x + 19, m<3 = 4y – 25, m<4 = 3y – 9, and arc PQ is congruent to arc RS. Find m<1, m<2, m<3, and m<4. Q R m<3and m<4 Since arc PQ is congruent to arc RS, <3 and <4 intercept congruent arcs. Thus, m<3 = m<4. m<3 = m<4 4y – 25 = 3y – 9 y – 25 = -9 y = 16 m<3 = 4y – 25 m<3 = 4(16) – 25 m<3 = 64 – 25 m<3 = 39 = m<4 . X 4 3 S P 1 2 T

  10. Ex 5: Find the value of x Find x. m<STQ = ½ arc ST 2x + 1 = ½ (96) 2x + 1 = 48 2x = 47 x = 23.5 Find x. <PHM and <ATP intercept the same arc so m<PHM = m<ATP. m<PHM = m<ATP x = 2x – 3 -x = -3 x = 3 S 96 (2x + 1) T Q P A X (2x – 3) H T

  11. Ex 6: In circle P, arc EN = 66, m<GPM = 89, and GN is a diameter. Find each measure. The measure of arc GM. Since <GPM is a central angle, arc GM is the same measure. Measure of arc GM = 89. B) The measure of arc NM. Since GN is a diameter, arc GMN is 180. arc GMN = arc GM + arc NM 180 = 89 + arc NM 91 = arc NM m<EGN Since <EGN is an inscribed angle, it is one half the measure of the intercepted arc. m<EGN = ½ (arc EN) m<EGN = ½ (66) m<EGN = 33 G E . P P N M

More Related