Circle theorems

Circle Theorem 1 - Angle at the Centre

[justify][b]1) [/b]The [color=#0000ff]central angle A [/color]and the [color=#0000ff]inscribed angle E[/color] are related to the[color=#76a5af] [/color][color=#93c47d]green arc CBD.[/color][color=#00ff00] [/color]Move the points C, D, and E to explore the geogebra and then answer the questions below.[/justify][b] a) [/b]Write down what you notice about the relationship between the two angles.[br][br][b] b) [/b] Conjecture (idea for what the rule might be):[br][br][b] c)[/b] Justification (using rules you already know):

Circle Theorem 2: Angles in a semicircle

[justify][b]2) [/b]The [color=#0000ff]diameter DB[/color] divides the[color=#0000ff] circumference into two equal parts (semicircles).[/color] The [color=#0000ff]inscribed angle C [/color] is subtended (related) by the [color=#ff0000]red arc DEB.[/color] Move point C around the circumference and then move the diameter DB as well. Explore it and answer the questions below.[/justify][b] a) [/b]Write down what you notice about the relationship between angle C and arc BD.[br][br][b] b) [/b] Conjecture (idea for what the rule might be):[br][br][b] c)[/b] Justification (using rules you already know):

Circle Theorem 3 - Inscribed angles subtended by the same arc

[b]3) [/b]The[color=#0000ff] Inscribed angles B and D[/color] are subtended by the same [color=#0000ff]blue arc AEC[/color]. Move the points around the circumference and then answer the questions below.[br][b][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):

Circle Theorem 4 - Cyclic Quadrilateral

[b]4) [/b]The [color=#0000ff][b]quadrilateral ABCD[/b][/color] is inscribed into the circle. Move the points around the circle and answer the questions below.[b][br][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):

Circle Theorem 5 - Alternate Segment Theorem

[b]5) [/b]Observe the angles between the[color=#0000ff] chords BA and CA[/color] and a[color=#0000ff] tangent[/color] passing through[color=#0000ff] point A.[/color] Move the points A, B, and C around the circumference and answer the questions below.[br][b][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):

Circle Theorem 6 - Tangent on a Circle (Tangent and Radius)

[b]6) [/b]The[color=#0000ff] blue line[/color] is tangent to the circle in the[color=#0000ff] point B[/color] and [color=#0000ff]segment AB[/color] is the radius of the circle. Move the point B around the circle and answer the questions below.[b][br][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):

Circle Theorem 7 - Two Tangent Theorem

[b]7) [/b] Observe the two [color=#ff0000]red tangents[/color] drawn from an external point to the circle passing by[color=#ff0000] points A and B. [/color]Move the slider and answer the questions below.[br][b][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):

Circle theorem 8 - Radius and Chord

[b]8) [/b]Observe the [color=#ff0000]chord CD[/color] and the [color=#ff0000]radius AF.[/color] Move the points C and D around the circumference.[br][b][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):

Extra - Circle Theorem 9 : Intersecting Chord

[b]9) [/b]Observe the two chords [color=#ff0000]CD and EF[/color]. Move the points C, D, E, and F around the circumference and answer the questions below.[br][b][br] a) [/b] What do you notice?[br][br][b] b) [/b]Conjecture (idea for what the rule might be):[br][br][b] c) [/b]Justification (using rules you already know):