The three theorems for the intercepted arcs to the angle of two tangents, two secants or 1 tangent and 1 secant are summarized by the pictures
below. If you look at each theorem, you really only need to remember ONE formula.
The formula:the angle formed by the intersection of 2 tangents, 2 secants or 1 tangent and 1 secant outside the circle equals half the difference of the intercepted arcs! Therefore to find this angle (angle K in the examples below) , all that you have to do is take the far intercepted arc and near the smaller intercepted arc and then divide that number by two!
That's why we call this the Far Arc Near Arc theorem , (sometimes abbreviated Farc - Narc)
Far Arc − Near Arc Formula
All of the formulas on this page can be thought of in terms of a "far arc" and a "near arc". The angle formed outside of the circle is always equal to the the far arc minus the near arc divided by 2.
It is worth noting that the regions of the circle that are not intercepted arcs do not factor into this formula and that for case III (two tangents) that every portion of the circle is used up because the intercepted arcs divide the entire circle into two parts.
Remember that this theorem only used the intercepted arcs. Therefore, the red arc in the picture below is not used in this formula.
Practice Problems
Problem 1)
Use the theorem for the intersection of a tangent and a secant to find the measure of the angle formed
by the intersection of the tangent and the secant.
Theorem: The measure of an angle formed by a two tangents drawn from a point OUTSIDE the circle is half the the difference of the intercepted arcs
This theorem differs from the other two on this page in that every part of the circle is included in the intercepted arcs. Since both of the lines are tangents, they touch the circle in one point and therefore they do not 'cut off' any parts of the circle.
Problem 3)
What is t he measure of x in the picture on the left. (Both lines in the picture are tangent to the circle)
Answer
$
x = \frac{1}{2} (205-155)
\\
x = \frac{1}{2} (50) = 25
$
The measure of an angle formed by a two secants drawn from a point outside the circle is half the the difference of the intercepted arcs. (far arc - near arc)
In the picture below, the measure of angle x is half the difference of the arcs intercepted by the two secants
Remember that this theorem only makes use of the intercepted arcs. Therefore, the red arcs in the picture below are not used in this theorem's formula.
Problem 5)
Two secants extend from the same point and intersect the circle as shown in the diagram below. What is the value of x?
Answer
$ x = \frac{1}{2}(140-50)
\\
x = \frac{1}{2}(90)
\\
x = 45^{\circ}
$
Brain Teaser
Use your knowledge of the theorems on this page to determine at whether point C or point D is where the bottom segment intersects the circle. In other words, is point D tangent to the circle?
Answer
Since $$ \frac{1}{2}(113- 45) \ne 35. $$ The segment is not tangent to the circle at C.
However, $$\frac{1}{2}(115- 45) = 35 $$ so the segment intersects point D. (the 115 represents 113 + 2 which is the sum of arc ABC + arc CD)
