Relations among Arcs, Central Angles and Inscribed Angles
Objective: The learner derives inductively the relations among arcs, central angles, and inscribed angles.
Name and Section:
Instructions: Observe the following diagrams, then answer the following questions. (You can manipulate the following illustration for further observation)
[Diagram #1]
Guide Questions:
1. Name the central angle and its intercepted arc.
2. What is the measure of the given central angle and its intercepted arc?
3. Now, move point C along the circumference of the circle clockwise to change the measure of the central angle. Observe the measure of the new central angles and the new intercepted arcs. Write your observation on the space below.
4. Complete the statement below:[br]The measure of the central angle is ________ to the measure of its _______________.
[Diagram #2]
Guide Questions:
1. Name the inscribed angle and its intercepted arc.
2. What is the measure of the inscribed angle and its intercepted arc?
3. Now, move point C along the circumference of the circle to change the measure of the inscribed angle and its intercepted arc. Write your observation in the space below.[br]
4. Complete the statement below: [br]The measure of the inscribed angle is ______________ the measure of its _____________.
[Diagram #3]
Guide Questions:
1. Name the 2 inscribed angles and their intercepted arc.[br][br]
2. What are the measures of the 2 inscribed angles?
3. Now, move points C and/or D along the circumference of the circle to change the measure of the inscribed angles and their intercepted arc. Write your observation in the space below.[br]
4. Complete the statement below: [br]If the _______ inscribed angles of a circle intercept the ___________ arc, then the angles are _________________.
[Diagram #4]
Guide Questions:
1. Name the inscribed angle and its intercepted arc.
2. What is the measure of the inscribed angle?[br]
3. What do you observe with the intercepted arc?[br]
4. Now, move points A and/or P along the circumference of the circle to change the position of the inscribed angle and its intercepted arc. Write your observation in the space below.[br]
5. Complete the statement below: [br]If an inscribed angle intercepts a _________________, then the angle is a __________________.
[Diagram #5]
Guide Questions:
1. Move points along the circumference of the circle to change the angle measures of the inscribed quadrilateral. Drag the slide me button for further observation. Write your conclusion in the space below.[br]
2. Complete the statement below:[br]In a cyclic quadrilateral, ________________ angles are ____________________.
[Diagram #6] Observe the diagram below. Drag points A and B to change the angle measure of angle O and the arc lengths of arcs CD and AB.
Guide Questions:
1. Name the 2 central angles and their intercepted arc.[br]
2. What are the measures of the 2 central angles?[br]
3. Write the length of the arcs.
4. Now, move point B along the circumference of the circle to change the measure of the central angles and their intercepted arcs' length. Write your observation in the space below.[br]
5. Complete the statement below: [br]Two arcs that have the ___________ measure, are not necessarily _______________ arcs.
Circles and Central Angles 2 Solving
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Angles ∠BAC and ∠DAE are central angles for circle A. Slider ca1 controls the size of ∠BAC. Slider ca2 controls the size of ∠DAE. [br][br]1. If an angle is larger, its arc is __________________. [br]2. Make ∠DAE and ∠BAC into congruent angles. Drag B and D so the arcs are aligned (on top of one another). If the angles are congruent, how are the arcs related? Try changing the angle lengths to make sure. [br]3. Can you make both arcs cover the entire circle without overlapping? What is the total of the central angles measures if they do? Is this always the case?[br]4. Can you make both arcs cover the entire circle without overlapping and have the central angles equal? What kind of angles and arcs do you form?[br]5. Make EB a diameter and give ∠DAE and ∠BAC two different measurements < 180°. Now solve for the missing angles ∠DAB and ∠CAE. Explain your process.