10.04 Inscribed angles and polygons

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.

Exploration

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

1. What relationship exists between the measure of the inscribed angle and its intercepted arc?
2. 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.

The measure of an angle is half the measure of the intercepted arc, so $a=\frac{1}{2}b$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, $\angle ADB$∠ADB and $\angle BCA$∠BCA intercept the same arc. Therefore, $\angle ADB\cong\angle BCA$∠ADB≅∠BCA.

Angles intercepting the same arc are congruent.

Inscribed polygons

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.

Exploration

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∠ACB=90° if and only if $\overline{AB}$AB is a diameter of the circle.

Since $\overline{AB}$AB is a diameter of circle $O$O, $\angle C$∠C is a right angle.

 

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 $ABCD$ABCD 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°.

In quadrilateral $ABCD$ABCD, opposite angles are supplementary. 

Practice questions

Question 1

Question 2

Question 3