Introduction: A
circle is all points equidistant from one
point called the
center of the circle. Segments drawn within, through, or tangent to the
circle create angles which we can define and measure.
Important facts:
 The measure of a central angle is the same as the measure of the intercepted arc.
 The measure of an inscribed angle is half the measure of the intercepted arc.
 A segment connecting two points on a circle is called a chord.
 A line passing through two points on a circle is called a secant.
 A line external to a circle, passing through one point on the circle, is a tangent.
The Lesson:
We show
circle O below in
Figure A.
Points A, C, B, D, and P are on the circle.
The segments AB and CD are chords.
We have also drawn segments AC and BD to form triangles ACQ and DBQ.
In Figure B we show just the chords AB and CD and the triangles they form: ACQ and DBQ.
Figure A

Figure B



ANGLES formed by chords in a circle:
Figure A

Using Figure A, since angles 2 and 3 are inscribed angles
The theorem from geometry states that an exterior angle of a triangle is equal in measure to the sum of the two remote interior angles. Therefore

SEGMENTS formed by intersecting chords in a circle:
Figure B

Using Figure B, we can examine the triangles ACQ and DBQ. Because they are inscribed angles intercepting the same arcs, we have This means that all three angles of the triangles ACQ and DBQ are equal in the correspondence: A~D, C~B, and of course Q~Q
We now know that DACQ ~ DDBQ, ab = cd

In Summary:
 The angle formed by intersecting chords is equal to ½ the sum of the intercepted arcs.
 The products of the segments formed by intersecting chords are equal.
Let's Practice:
 In the diagram shown below, circle O is given with ^{} and ^{}. What is the measure of angle 1?
Since
angle 1 is formed by intersecting chords, we have:
 The diagram given below shows circle O with chords AB and CD. CD is divided into segments of lengths x and 4x. AB is divided into segments of lengths 2x and 5. What is the measure of chord CD?
The products of the segments forming the chords are equal.
x(4x) = 2x(5)
4x^{2} = 10x
4x
^{2}  10x = 0
x(4x  10) = 0
x = 0 or 4x = 10
x = 5/2
We will not use x = 0 since
chord CD would also equal 0.
If we use x = 5/2,
chord CD would measure
CD = x + 4x
CD = 5/2 + 4(5/2)
CD = 25/2