12. Circles

Lesson

An inscribed angle is an angle with a vertex is on the circumference of a circle and legs that contain the chords of a circle. The arc that lies in the interior of the inscribed angle is called the intercepted arc.

In the applet below $\angle DFE$∠`D``F``E` is the inscribed angle and arc $DE$`D``E` is its intercepted arc. Move points $D$`D`, $E$`E`, and $F$`F` around the circle and try to answer the following questions.

- What relationship exists between the measure of the inscribed angle and its intercepted arc?
- What has to happen for $\angle F$∠
`F`to be a right angle?

From the applet, we can see that the inscribed angle is always half the measure of its intercepted arc. Since we can prove this is true, it's known as the inscribed angle theorem.

Inscribed angle theorem

If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc.

In the diagram below, $a=\frac{1}{2}b$`a`=12`b`.

In the applet above, we saw that if we move the point on the circumference of the circle, while keeping the arc the same, the measure of the inscribed angle remains the same. This is a corollary to the inscribed angle theorem.

Corollary

If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.

In the diagram below, $\angle ADB$∠`A``D``B` and $\angle BCA$∠`B``C``A` intercept the same arc. Therefore, $\angle ADB\cong\angle BCA$∠`A``D``B`≅∠`B``C``A`.

We say that a polygon is inscribed in a circle if all of its vertices are on the circumference of the circle. The polygon is sometimes referred to as an inscribed polygon.

Triangles and quadrilaterals that are inscribed in circles have special properties. First, let's think about the relationship between a semicircle and an inscribed right angle.

Suppose that an inscribed angle has a measure of $90^\circ$90°. Then it follows from the inscribed angle theorem that its intercepted arc is $2\times90^\circ=180^\circ$2×90°=180°. An arc measuring $180^\circ$180° is a semicircle. Therefore, an inscribed right angle intercepts a semicircle and the two chords form a diameter of the circle. We can prove the converse in a similar way.

Corollary

An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle.

For the diagram below, $m\angle ACB=90^\circ$`m`∠`A``C``B`=90° if and only if $\overline{AB}$`A``B` is a diameter of the circle.

The following theorem also follows from the inscribed angle theorem. As an exercise, see if you can prove its truth in a two-column or paragraph proof.

Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

For example, in quadrilateral $ABCD$`A``B``C``D` below, we know that $m\angle A+m\angle C=180^\circ$`m`∠`A`+`m`∠`C`=180° and $m\angle B+m\angle D=180^\circ$`m`∠`B`+`m`∠`D`=180°.

Solve for $x$`x`.

Find the value of $x$`x`.

In the diagram, $O$`O` is the center of the circle.

Solve for $p$

`p`.Solve for $q$

`q`.Solve for $r$

`r`.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.