Angle of Intersecting Chords Theorem

When we start drawing lines in circles, a few interesting things start to happen. One of the patterns we may notice is the creation of angles in the center of the circle. These angles follow very specific rules, and one of these is highlighted by the angles of intersecting chords theorem. But what is this theorem really about? Let's find out:

The angles of intersecting chords theorem explained

The angles of intersecting chords theorem states that:

• If two chords intersect inside a circle, then the measure of the angle formed is half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

Let's back up for a moment here and define a few terms:

• A chord is simply a line through a circle
• An arc is the segment of a circle's perimeter created by two intersecting lines
• Vertical angles are pairs of opposite angles formed by intersecting lines. They are across from each other

Visualizing the angles of intersecting chords theorem

It helps if we can visualize all of this taking shape. Consider the following circle:

As we can see, there are four angles formed by the two intersecting chords: PR and RS. There are also two arcs shaded in red: $\stackrel{⌢}{\mathrm{PQ}}$ and $\stackrel{⌢}{\mathrm{RS}}$ .

If we were to put the intersecting chords theorem into a formula, it would look something like this:

$\angle 1=\frac{1}{2}(\stackrel{⌢}{\mathrm{PQ}}+\stackrel{⌢}{\mathrm{RS}})\angle 2=\frac{1}{2}(\stackrel{⌢}{\mathrm{QR}}+\stackrel{⌢}{\mathrm{PS}})$

Remember, vertical angles are congruent -- which means that their measures are equal. This means that angle 1 is congruent to angle 3, and angle 2 is congruent to angle 4.

Working with the angles of intersecting chords theorem

Let's take another look at that circle:

If $\stackrel{⌢}{\mathrm{PQ}}=92$ degrees and $\stackrel{⌢}{\mathrm{RS}}=110$ degrees, then can we find the measure of angle 3?

We know that:

$\angle 3=\frac{1}{2}(\stackrel{⌢}{\mathrm{PQ}}+\stackrel{⌢}{\mathrm{RS}})$

Let's plug in our values:

$\frac{1}{2}(92+110)$

$=\frac{1}{2}\left(202\right)$

$=101$

Now we know that angle 3 equals 101 degrees!

Topics related to the Angle of Intersecting Chords Theorem

Chord

Circle

Arcs

Flashcards covering the Angle of Intersecting Chords Theorem

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Angle of Intersecting Chords Theorem

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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