Tangent , Secant, arcs and angles of a Circle

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 angle formed by the intersection of 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 (the shaded angle in the examples below) just take the larger intercepted arc and subtract the smaller intercepted arcand then divide that difference by two!

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.

The Technical Formulas

Just show me the easy way to remember the formulas

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.

Theorem: The measure of an angle formed by a secant and a tangent drawn from a point OUTSIDE the circle is half the the difference of the intercepted arcs.

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

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.

Answer

Answer

68= ½ (158 - )
2 × 68 = 158 -
136= 158 -
22 =

Only one of the two circles BELOW includes the intersection of a tangent and a secant.
Can you figure out which one?

Answer

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.

Rememer that this theorem only makes use of the interecepted arcs. Therefore, the red arcs in the picture below are not used in this theorem's formula.

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 = ½(140-50)
x = ½(90)
x = 45°

Find the measure of angle k, where the two secant segments intersect.

  answer  

K = ½(90-30)
K = ½(60)
K = 30°

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 the bottom segment, FD or FC?


  answer  

Since ½(113- 45) ≠35. The segment does not intersect the circle at C.;
However, ½(115- 45) = 35 so the segment intersects point D. (the 115 represents 113 + 2 which is the sum of arc ABC + arc CD)

In the circle below, AE is a diameter. Use the formula to calculate the measure of angle K .

Answer

Since AE is a diameter its arc measuers 180°. You can use this fact and the ratio to set up an equation and get the values of both arcs. See work in the picture below. From there just employ the formula from this page.