University:
University of Houston
Course:
MATH 3335 | Vector Analysis
Academic year:

2024
Views:

234

Pages:

11
Author:

Tristin Moyer

Example 3 In O of Figure 6.11, bisects AOD. Figure 6.11 Using the measures indicated, find: a) b) c) e) f) g) whether d) h) The measure of the reflex angle that intercepts 24 Example 3 – Solution a) 105° b) 70° c) 105° d) 150°, from 360 – (105 + 70 + 35) e) 75° because the corresponding central angle ( AOE) is the result of bisecting AOD, which was found to be 150° 25 Example 3 – Solution f) 285° (from 360 – 75, the measure of cont’d ) g) The arcs are congruent because both measure 75° and are arcs in the same circle. h) 210° (from 105° + 70° + 35°) 26 Angle and Arc Relationships in the Circle In Figure 6.11, note that (or ). is the major arc . , we Figure 6.11 Because the union of also see that and 27 Angle and Arc Relationships in the Circle With the understanding that and generalize the relationship as follows. do not overlap, we Postulate 17 (Arc-Addition Postulate) If and intersect only at point B, then Given points A, B, and C on O as shown in Figure 6.12(a), suppose that radii , and are drawn. Figure 6.12(a) 28 Angle and Arc Relationships in the Circle Just as by the Angle-Addition Postulate, it follows that In the statement of the Arc-Addition Postulate, the reason for writing (rather than ) is that the arc with endpoints at A and C could be a major arc. The Arc-Addition Postulate can easily be extended to include more than two arcs. 29 Angle and Arc Relationships in the Circle In Figure 6.12(b), Figure 6.12(b) If alternately, , then point S is the midpoint of is bisected at point S. ; 30 Angle and Arc Relationships in the Circle Definition An inscribed angle of a circle is an angle whose vertex is a point on the circle and whose sides are chords of the circle. Theorem 6.1.2 The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc. 31 Angle and Arc Relationships in the Circle Theorem 6.1.3 In a circle (or in congruent circles), congruent minor arcs have congruent central angles. Theorem 6.1.4 In a circle (or in congruent circles), congruent central angles have congruent arcs. Theorem 6.1.5 In a circle (or in congruent circles), congruent chords have congruent minor (major) arcs. 32 Angle and Arc Relationships in the Circle Theorem 6.1.6 In a circle (or in congruent circles), congruent arcs have congruent chords. Theorem 6.1.7 Chords that are at the same distance from the center of a circle are congruent. 33 Angle and Arc Relationships in the Circle Theorem 6.1.8 Congruent chords are located at the same distance from the center of a circle. Theorem 6.1.9 An angle inscribed in a semicircle is a right angle. Theorem 6.1.10 If two inscribed angles intercept the same arc, then these angles are congruent. 34

Circles and Related Segments and Angles 3

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