Arc and Angle Measures inside and outside a circle
1. Vertex at the Center
1. Explore Angles Inside and Outside Circles
2. Central Angles and Interior Angles
2. Vertex inside the circle (NOT at Center)
1. Angles Inside the Circle Theorem
2. Angle inside a circle
3. Secants Angles Inside
4. Arc and Angle Inside a Circle
3. Vertex ON the circle or Inscribed Angle
4. Vertex OUTSIDE the circle
1. IM2.7 T8 Angles formed by Secants (outside)
2. Vertex outside the circle (Secant Angles)
3. Angle Formed by a Tangent and a Secant Intersecting at a Point Outside the Circle
4. Angle Formed by a Tangent and a Secant Intersecting at a Point Outside the Circle

0

Arc and Angle Measures inside and outside a circle

Ella Platt, Apr 16, 2026

Angles and arcs in circles

Vertex at the Center
1. Explore Angles Inside and Outside Circles
2. Central Angles and Interior Angles
Vertex inside the circle (NOT at Center)
1. Angles Inside the Circle Theorem
2. Angle inside a circle
3. Secants Angles Inside
4. Arc and Angle Inside a Circle
Vertex ON the circle or Inscribed Angle
Vertex OUTSIDE the circle
1. IM2.7 T8 Angles formed by Secants (outside)
2. Vertex outside the circle (Secant Angles)
3. Angle Formed by a Tangent and a Secant Intersecting at a Point Outside the Circle
4. Angle Formed by a Tangent and a Secant Intersecting at a Point Outside the Circle

Vertex at the Center

1. Explore Angles Inside and Outside Circles
2. Central Angles and Interior Angles

Explore Angles Inside and Outside Circles

[color=#ff00ff][size=100][size=150]*****Use this animation to confirm what you discovered about the relationship of angles inside and outside a circle and the 2 arcs intercepted by the sides of the angle.*****[br][br][/size][/size][/color][i]Watch the formula at the bottom of the applet and notice how it changes when the vertex of the angle is inside the circle vs. outside the circle.[/i][br][br]*Move Point C or D to change the size of Arc1 and Arc2. [br][br]*Move Point E to change the location of the vertex of the angle or the size of Arc 2.

1.2.

2.

Vertex inside the circle (NOT at Center)

1. Angles Inside the Circle Theorem
2. Angle inside a circle
3. Secants Angles Inside
4. Arc and Angle Inside a Circle

2.1.

Angles Inside the Circle Theorem

[color=#000000]In the applet below, the [/color][b]gray angle[/b][color=#000000] is said to be an [/color][b]angle formed by 2 chords of a circle[/b][color=#000000].[/color]

2.2.

2.3.

2.4.

3.

Vertex ON the circle or Inscribed Angle

This chapter does not contain any resources yet.

4.

Vertex OUTSIDE the circle

1. IM2.7 T8 Angles formed by Secants (outside)
2. Vertex outside the circle (Secant Angles)
3. Angle Formed by a Tangent and a Secant Intersecting at a Point Outside the Circle
4. Angle Formed by a Tangent and a Secant Intersecting at a Point Outside the Circle

4.1.

IM2.7 T8 Angles formed by Secants (outside)

In the last task, you found a relationship between the angle formed by two tangent lines that intersect outside of a circle and the arc they intercept. This task will extend that reasoning to secant lines.[br][br][color=#000000]Interact with the applet below for a few minutes. Then answer the questions that follow. [br][br][/color][color=#0000ff]Be sure to change the locations of the[/color][color=#000000] [b]BIG POINTS[/b] [/color][color=#0000ff]each time [i]before[/i] you slide the slider. [/color]

[br][color=#000000]1) Suppose the pink arc measures 200 degrees and the green arc measures 50 degrees. What would the measure of the blue angle be? [br][br]2) Move the pink points so that only 1 of the rays becomes tangent to the circle (while keeping the other ray a secant ray.) Answer question #1 again within this context. [br][br]3) Move the pink points so that both rays become tangent to the circle. Suppose, in this case, the entire pink arc measures 210 degrees. What would the measure of the green arc be? What would the measure of the blue angle be? [br][br]4) Next, move the blue point as close to the circle as possible so that the green arc almost disappears. (It won't disappear entirely). Keep the blue point on the circle. Now slowly re-slide the slider again. What previously learned theorem do these transformations reveal? [br][br]5) Suppose 2 secant rays (drawn from a point outside a circle) intersect the circle above so that the blue angle measures 60 degrees and the entire pink arc measures 200 degrees. If this is the case, what would the measure of the entire green arc be? [br][br]6) Answer question number 5 again, this time in the case of a secant and a tangent.[/color]

0