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Arcs and Central Angles
Central angles are angles whose vertex is the center of a circle. A central angle separates a circle into two arcs: a minor arc and a major arc. The measure of a minor arc is equal to the measure of its central angle. The measure of a major arc is equal to 360 degrees minus the measure of the minor arc. The measure of an arc formed by two adjacent arcs is the sum of the measures of the individual arcs. If two minor arcs in the same or congruent circles are congruent, then the corresponding chords are also congruent.
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Arcs and Central Angles
- 1. ARCS AND CENTRALSANGLES
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- 3. Semicircle – apart of a circle from one end point of the diameter to the other endpoint. It is half of the circle. Definition of Parts of a Circle The degree measure of semicircle 180º.
- 4. Central angle –is an angle whose vertex is the center of the circle. - Central angle separates a circle into two arcs called minor arcs and major arcs. ∠AOB
- 5. Central Angle ∠AOB Minor Arc AB Minorarc AB - Major Arc The degree measure of a minor arc - is equal to the measure of the central angle. Is smaller than a semicircle.
- 6. Example: m∠AOB = 45º, 45º AB = 45º Minor Arc AB 45º m
- 7. Central Angle ∠AOB Minor Arc AB Majorarc ACB - Is larger than a semi circle. Major Arc ACB The degree measure of a major arc - is equal to 360 minus the degree measure of its related minor arc.
- 8. Example: m ACB= 360ºm AB = 45º Find the measure of major arc m ACB= 360º - 45º m ACB= 315º
- 9. Arc Addition Postulate Themeasure of an arc formed by two adjacent, non overlapping arc is the sum of the measures of two arcs.
- 11. b. Find mADC a. Find m ABC m ABC = m AB + m BC = 55º + 70º = 125º m ADC = 360º - 125º =235º m ADC = m AD + m DC OR = 195º + 40º = 235º
- 12. Congruent Arcs In congruentcircles, arcs which have the same measure are congruent arcs Theorem If two minor arcs of a circle or of congruent circles are congruent, then the corresponding chords are congruent.
- 14. STATEMENTS REASONS 1. CircleO with AB ≅ CD 2. mAB = mCD 3. m∠AOB = mAB 4. OA ≅ OC OB ≅ OD
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