An inscribed angle is an angle that has its point (or vertex) on a circle, and its sides are made of lines (called chords) that go from one side of the circle to the other. One cool thing about inscribed angles is how they relate to the arcs (the curved parts of circles) they touch. In fact, the size of an inscribed angle is always half the size of the arc it is associated with.

Definition and Formula

1. Inscribed Angle: Imagine an angle that’s named $\angle ABC$. Here, point $B$ is on the circle, while lines $AB$ and $BC$ are the sides of the angle. The part of the circle that is inside the angle is called arc $AC$.

2. Measure of Inscribed Angle: We can express the size of the inscribed angle like this: $m\angle ABC = \frac{1}{2} m\overset{\frown}{AC}$ In this formula, $m\angle ABC$ shows the measure of the inscribed angle, and $m\overset{\frown}{AC}$ shows the size of the arc $AC$.

Effect of Changing the Arc

When the size of arc $AC$ changes, the size of the inscribed angle $ABC$ changes too. Here are some examples:

• Increasing the Arc:

  • If arc $AC$ gets bigger, its measurement in degrees also goes up to a maximum of $180^\circ$ (which is half a circle).
  • Because of this increase, the size of the inscribed angle also grows. For example, if arc $AC$ is $80^\circ$, then the inscribed angle $ABC$ would be: $m\angle ABC = \frac{1}{2} \times 80^\circ = 40^\circ$

• Decreasing the Arc:

  • On the other hand, if arc $AC$ gets smaller, the inscribed angle also gets smaller.
  • So, if arc $AC$ now measures $40^\circ$, then the inscribed angle $ABC$ would measure: $m\angle ABC = \frac{1}{2} \times 40^\circ = 20^\circ$

Summary of Relationships

1. General Observations:

   • The way inscribed angles change is directly linked to how the arcs change.
   • Inscribed angles can measure from $0^\circ$ to $90^\circ$ (acute angles), from $90^\circ$ to $180^\circ$ (obtuse angles), and can even be $180^\circ$ when the arc is the full $360^\circ$ (the whole circle).

2. Special Cases:

   • If the arc measures $180^\circ$, the inscribed angle will be $90^\circ$. This is especially important because it relates to the idea that a triangle inside a semicircle always will have a right angle (this idea is known as Thales' theorem).
   • If two inscribed angles share the same arc, they will have the same angle measure.

Conclusion

The relationship between inscribed angles and the arcs they touch is really important for understanding circles in geometry. Knowing how the size of an inscribed angle changes when the size of the arc changes helps students deal with more complicated circle problems, including those involving tangents and secants. This concept isn't just for the classroom—it's also useful in real life, like in design, architecture, and engineering, where circles are often used!