Now let's extend our knowledge of Thales' theorem to a new case.

How are the values of $x$ and $y$ related in the figure below?

The green angles are inscribed angles and the orange angle is a central angle.

We can see that $y$ is larger than $x,$ but we want to determine how much larger. If we know that the purple angle measures $180^\circ - y,$ then what can we say about how $x$ and $y$ compare?

Using the diagram below and our previous theorems, we know that $b = 2a$ and that $y = 2x.$

Is it always, sometimes, or never true that $b+y = 2(a+x) ?$

Congratulations! You've now proven the inscribed angles theorem!

As we saw in the last few problems, central angles and inscribed angles have a special relationship:

**Whenever a central angle and an inscribed angle share the same endpoints, the measure of the central angle will always be twice the measure of the inscribed angle.**

This property is known as the **inscribed angle theorem**, and over the last several problems, you proved it!

The image above illustrates three cases that look different, but all three are instances when the inscribed angles theorem would apply.

A quick reminder of the relevant definitions:

An **inscribed angle** is an angle on the interior of a circle that forms where two chords meet:
A **central angle** is an angle on the interior of a circle whose vertex is the center of the circle and whose legs (sides) are radii:

**Applying the Inscribed Angle Theorem:**

True or False?

From the diagram below, we can definitely conclude $x = y .$

What is the value of $x\,?$

**Applying the Inscribed Angle Theorem:**

What is the value of $r + b\,?$

So far, you've learned about angles in circles, Thales' Theorem, and the Inscribed Angle Theorem.

In the main course, you'll take on case #3 of the Inscribed Angle Theorem and then discover many more geometric theorems, including one of the most astonishing of all, Euler's line.

×

Problem Loading...

Note Loading...

Set Loading...