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"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 3
Knowledge review
1. A circle is an axially symmetrical figure.
The axis of symmetry of a circle is any straight line passing through the center of the circle. It has countless axes of symmetry.
2. A circle is also a centrally symmetrical figure.
Its center of symmetry is the center of the circle.
3. The angle whose vertex is at the center of the circle is called the central angle.
4. Theorem: In congruent circles or equal circles, the arcs subtended by equal central angles are equal, and the chords subtended by equal central angles are equal.
5. Theorem: In congruent circles or equal circles, if one set of quantities in two central angles, two arcs, and two chords is equal, then the remaining sets of quantities corresponding to them are equal.
Perpendicular diameter theorem
The diameter perpendicular to the chord bisects the chord and bisects the two arcs subtended by the chord.
inference
(1) The diameter that bisects the chord (not the diameter) is perpendicular to the chord and bisects the two arcs subtended by the chord
(2) The perpendicular bisector of the chord passes through the center of the circle and bisects the two arcs subtended by the chord.
(3) Bisect the diameter of an arc, perpendicularly bisect the chord subtended by the arc, and bisect the other arc subtended by the chord.
Proposition (1): The diameter that bisects the chord (not the diameter) is perpendicular to the chord and bisects the two arcs subtended by the chord.
∵CD is the diameter, AB is the chord, and CD bisects AB
∴CD⊥AB, AD=BD, AC=BC
Proposition (2): The perpendicular bisector of the string passes through the center of the circle and bisects the two arcs subtended by the string.
∵ AB is a string, CD bisects AB, CD ⊥AB,
∴ CD is the diameter, AD=BD, AC=BC
Proposition (3): Bisect the diameter of an arc, perpendicularly bisect the chord subtended by the arc, and bisect the other arc subtended by the chord.
∵ CD is the diameter, AB is the chord, and AD=BD (AC=BC)
∴ CD bisects AB, AC=BC (AD=BD) CD ⊥AB
Do it
As shown in the figure, ∠AOB=80°.
(1) Please draw several circumferential angles that AB � corresponds to. What is the relationship between these circumferential angles? Please communicate with your companions.
(2) What is the relationship between these circumferential angles and the size of the central angle ∠AOB? How did you find out? Communicate with peers.
Discuss: If the degree of ∠AOB is changed, does the above conclusion still hold?
Circle Angle Theorem
The circumferential angle subtended by an arc is equal to half of the central angle subtended by it.
Determine the circumferential angle by analogy with the central angle of a circle
In congruent or equal circles, equal arcs subtend equal central angles.
What is the relationship between the angles subtended by equal arcs in congruent or equal circles?
In order to solve this problem, we first explore the relationship between the circumferential angle and the central angle of an arc.
Ask students to identify a minor arc on the circle and draw the central angle and circumferential angle it corresponds to.
Class summary
1. Definition of circumferential angle: An angle whose vertex is on a circle and both sides intersect the circle is called a circumferential angle.
2. Circumferential angle theorem: The circumferential angle subtended by an arc is equal to half of the central angle subtended by it.
3. The corollary of the circular angle theorem: the circular angles subtended by the same arc (equal arcs) are equal.
4. In congruent circles or equal circles, equal arcs subtended by equal circumferential angles are equal.
5. In the same circle or equal circles, the arcs subtended by equal chords are not necessarily equal.
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"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 5:
"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 5 Knowledge Review Circumferential Angle: The vertex is on the circle, and its two sides have another intersection with the circle. An angle like this is called a circumferential angle. Circumferential Angle Theorem The circumference of a circle subtended by an arc An angle is equal to half of the central angle of the circle it subtends. Health...
"The Relationship between the Circumferential Angle and the Central Angle" Circle PPT Courseware 4:
"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 4 Circumferential Angle Definition: An angle whose vertex is on a circle and both sides intersect with the circle is called a circumferential angle. Features: ① The vertex of the angle is on the circle. ② Both sides of the angle intersect with the circle Intersect. Hands on the relationship between the circumferential angle and the central angle of the circle subtended by the same arc..
"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 2:
"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 2 1. Review 1. What is a circumferential angle? The vertex is on the circle and the angle where the two sides intersect the circle is called the circumferential angle. 2. Fill in the blanks: ⑴The _______ subtended by an arc is equal to half of the degree _________ subtended by it. ⑵Circumference..