Circles, Angle Measures, Arcs, Central & Inscribed Angles, Tangents, Secants & Chords - Geometry
The Organic Chemistry Tutor・20 minutes read
In circle geometry, various angles and arcs are related, such as central angles, inscribed angles, secant angles, and tangent angles, with specific formulas and properties defining their measurements and relationships. Understanding these angle and arc relationships is crucial for solving problems and determining the measures of unknown angles within a circle.
Insights
- Inscribed angles have their vertex on the circle and are half the measure of the intercepted arc; for example, angle A is 20 degrees when the intercepted arc is 40 degrees.
- The sum of arcs AC and DE determines the measure of angle ABC, with the algebraic solution leading to x being 8 and arc AC measuring 90 degrees.
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Recent questions
What is the relationship between central angle and arc measure?
The central angle has its vertex at the center of the circle, with its measure equal to the measure of the intercepted arc. For example, if Angle ACB is 50 degrees, then Arc AB will also be 50 degrees.
How is an inscribed angle defined?
An inscribed angle has its vertex on the circle, with its measure equal to half the measure of the intercepted arc. For instance, if Angle ABC is 30 degrees, then Arc AC will be 60 degrees.
What is a tangent chord angle?
A tangent chord angle is formed by a tangent segment and a chord, with the angle measure equal to half the measure of the intercepted arc. If Angle ABC is 25 degrees, then Arc AB will be 50 degrees.
How is a secant angle calculated?
The secant angle formula states that the angle measure is half the difference between the two intercepted arcs. For example, if Angle B is 25 degrees, then the measure of Arc DC will be 70 degrees.
How is the measure of a chord-chord angle determined?
A chord-chord angle is half the sum of the two intercepted arcs. For instance, if Angle AFE is 100 degrees, then the sum of arcs AE and BD will be 200 degrees, leading to Angle C being 30 degrees.
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Summary
00:00
Circle Angle Formulas and Relationships
- Central angle has vertex at circle center; Angle ACB = 50 degrees, Arc AB = 50 degrees.
- Inscribed angle has vertex on circle; Angle ABC = 30 degrees, Arc AC = 60 degrees.
- Tangent chord angle formed by tangent segment and chord; Angle ABC = 25 degrees, Arc AB = 50 degrees.
- Chord chord angle formed by two intersecting chords; Angle ABC = 80 degrees, Arc AC = 100 degrees, Arc DE = 60 degrees.
- Measure of Arc AC = 40 degrees; Measure of Arc DE = 110 degrees; Measure of Angle EBC = 100 degrees.
- Measure of Arc AD = 105 degrees; Measure of Arc CE = 125 degrees.
- Secant angle formula: Angle B = 25 degrees.
- Secant tangent angle formula: Arc DC = 70 degrees, Arc AD = 160 degrees.
- Tangent tangent angle formula: Arc AC = 140 degrees, Angle B = 40 degrees.
- Inscribed angle Angle A = 20 degrees; Inscribed angle Angle C = 20 degrees.
22:32
Circle Geometry: Angles and Arcs Explained
- Angle a is the inscribed angle with an intercepted arc, making it half of the intercepted arc, which is 30 degrees.
- D is the center of the circle, making AC the diameter, forming a semicircle with an arc measure of 180 degrees.
- B is the inscribed angle for arc AC, making it 90 degrees, half of 180.
- A triangle formed across a diameter is always a right triangle, with angle C calculated as 60 degrees.
- Arc AC is 180 degrees, with arc AB being twice that value at 120 degrees, totaling 360 degrees around the circle.
- Angle x is calculated as 60 degrees, as it shares the same arc as angle D, making them congruent.
- Arc BE is 120 degrees, equal to 2x, leading to x being 60 degrees.
- The measure of angle ABC is half the sum of arcs AC and DE, resulting in an algebraic solution to find x as 8, making arc AC 90 degrees.
- Angle AFE, a chord-chord angle, is calculated as 100 degrees, half the sum of arcs AE and BD.
- Angle C is found to be 30 degrees, half the difference between arcs AE and BD, totaling 60 degrees.
