We will first explore the relationship among the angles and arcs formed by the intersection of a chord and a tangent line. Then, we will examine the relationship among the angles and arcs formed by the intersection of two chords.

Intersection of a Chord and a Tangent Line

Let's move the slider and examine how the measures of angles and arcs change.

As we can see, the measure of each angle formed is half the measure of its intercepted arc.

m ∠ 1 = \frac{1}{2} m A B and m ∠ 2 = \frac{1}{2} m B C A

Intersection of Two Chords

Let's construct two intersecting chords in a circle and find the measures of the angles and arcs formed.

We see that the measure of

∠ 1

equals half the sum of the measures of the arcs intercepted by

∠ 1

and its vertical angle.

m ∠ 1 = \frac{1}{2} (m A B + m C D)

Now, let

∠ 2

be one the supplementary angles to

∠ 1 .

Let's investigate the relation between the measure of

∠ 2

and the measures of

A D

and

D C .

Again, we see that

m ∠ 2

is equal to half the sum of the measures of the arcs intercepted by

∠ 2

and its vertical angle.

m ∠ 2 = \frac{1}{2} (m B C + m D A)

As a result of the explorations, we can say that the measure of each angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Exercise

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