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Tangents, Secants, Arcs and Angles

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.

Far Arc near arc formula
Case I. Tangent and Secant

The measure of an angle formed by a secant and a tangent drawn from a point outside the circle is $$ \frac 1 2 $$ the difference of the intercepted arcs .

Farc - Narc Formula Tangent Secant
Picture of intersection of tangent and secant

Remember that this theorem only used the intercepted arcs . Therefore, the red arc in the picture below is not used in this formula.

Only Intercepted Arcs
Case II. 2 Secants

The measure of an angle formed by a 2 secants drawn from a point outside the circle is half the the difference of the intercepted arcs:

In the picture below, the measure of $$ \angle x$$ is $$ \frac 1 2 $$ the difference of the arcs intercepted by the two secants.

Picture of intersection of two secants
Far Arc - Near Arc Formula

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.

Only Intercepted Arcs
Case III. 2 Tangents

The measure of an angle formed by a two tangents drawn from a point outside the circle is $$\frac 1 2 $$ the the difference of the intercepted arcs .

Picture of intersection of two tangents
Picture of intersection of tangent and secant

In one way, this case seems to differ from the others-- because all 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.

Interactive app
$$ \overparen{\rm Far} = \class{data-angle-0}{35.92} \\ \overparen{\rm Near} = \class{data-angle-1}{89.84} \\ \angle{Outer} = \frac{\overparen{\rm Far} - \overparen{\rm Near}}{2} \\ = \frac{\class{data-angle-0}{035.92} - \class{data-angle-1}{89.84}}{2} \\ = \class{data-angle-outer}{26.96} ^{\circ} $$
Drag Points To Start Demonstration

Applet on its own page

Practice Problems

Problem 1

What is the measure of $$ \angle x $$?

Apply the formula . $$ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle x = \frac{1}{2} \left( \overparen{AC} - \overparen{CH} \right) \\ m \angle x = \frac{1}{2} \left( \overparen{178^{\circ}} - \overparen{102^{\circ}} \right) \\ m \angle x = 38^{\circ} $$
picture of intersection of secant and a tangent
Problem 2

What is the measure of $$ \overparen{\rm CH} $$?

Apply the formula . $$ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ 68 =\frac{1}{2}(158 - \overparen{\rm CH}) \\ 2 \cdot 68 = 2 \cdot \frac{1}{2}(158 - \overparen{\rm CH}) \\ 136 = 158 - \overparen{\rm CH} \\ 22 = \overparen{\rm CH} $$
Tangent Secant Diagram
Problem 3

Only one of the two circles below includes the intersection of a tangent and a secant.

Can you figure out which one?

Only Circle 1 on the left is consistent with the formula.

$$ \text{Circle 1} \\ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle 64^{\circ} = \frac{1}{2} \left( 164 ^{\circ} - 36^{\circ} \right) \\ m \angle 64^{\circ} = \frac{1}{2} \left( 128^{\circ} \right) \\ m \angle 63^{\circ} \red{ \ne} \frac{1}{2} \left( 128^{\circ} \right) $$
$$ \text{Circle 1} \\ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle 64^{\circ} = \frac{1}{2} \left( 164 ^{\circ} - 36^{\circ} \right) \\ m \angle 64^{\circ} = \frac{1}{2} \left( 128^{\circ} \right) \\ m \angle 64^{\circ} = \frac{1}{2} \left( 128^{\circ} \right) $$
Rule Brain Teaser
Problem 4

What is the measure of x in the picture on the left. (Both lines in the picture are tangent to the circle)

Apply the formula .

$$ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle x = \frac{1}{2} \left( \overparen{CAH} - \overparen{CH} \right) \\ m \angle x = \frac{1}{2} (205-155) \\ m \angle x = \frac{1}{2} (50) \\ m \angle x = 25^{\circ} $$

picture of two tangents to circle
Problem 5

What is the measure of $$\overparen{\rm CH}$$?

Apply the formula .

$$ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle x = \frac{1}{2} \left( \overparen{CAH} - \overparen{CH} \right) \\ 30 =\frac{1}{2}(210- \overparen{\rm CH}) \\ 2 \cdot 30= 2 \cdot \frac{1}{2}(210- \overparen{\rm CH}) \\ 2 \cdot 30= (210- \overparen{\rm CH}) \\ 60 = 210 - \overparen{\rm CH} \\ 150^{\circ} = \overparen{\rm CH}$$

Two Tangents
Problem 6

Two secants extend from the same point and intersect the circle as shown in the diagram below. What is the value of x?

Apply the formula .

$$ m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ m \angle x = \frac{1}{2} \left( \overparen{ABC} - \overparen{XYZ} \right) \\ m \angle x = \frac{1}{2}(140-50) \\ m \angle x = \frac{1}{2}(90) \\ m \angle x = 45^{\circ} $$

Diagram of intersection of two secants of a circle
Problem 7

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?

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)

Two Secants Brain Teaser

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