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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 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

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: $\stackrel{\u2322}{\mathrm{PQ}}$ and $\stackrel{\u2322}{\mathrm{RS}}$ .

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

$\angle 1=\frac{1}{2}(\stackrel{\u2322}{\mathrm{PQ}}+\stackrel{\u2322}{\mathrm{RS}})\angle 2=\frac{1}{2}(\stackrel{\u2322}{\mathrm{QR}}+\stackrel{\u2322}{\mathrm{PS}})$

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.

Let's take another look at that circle:

If $\stackrel{\u2322}{\mathrm{PQ}}=92$ degrees and $\stackrel{\u2322}{\mathrm{RS}}=110$ degrees, then can we find the measure of angle 3?

We know that:

$\angle 3=\frac{1}{2}(\stackrel{\u2322}{\mathrm{PQ}}+\stackrel{\u2322}{\mathrm{RS}})$

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!

Common Core: High School - Geometry Flashcards

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The angles of intersecting chords theorem is just one of many geometrical theorems to keep track of. A qualified tutor can provide your student with plenty of tips and tricks to memorize these theorems. Students can also ask plenty of questions in a 1-on-1 environment that they might not have a chance to raise in class time. Contact Varsity Tutors, and we'll match your student with a tutor.

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