Tangent , Secant, ars and angles of a Circle

Tangents, secants, their arcs and angles: theorems

Tangent and Secant from a point

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?

Two Tangents from Point

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.

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

x = ½ (205-155)
x = ½ (50) =25

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

30 ° = ½ (210 − )
60° =210 −
− 150°= −
150°=

Two intersecting Secants from a Point

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.

In the picture below, the measure of angle x is half the difference of the arcs intercepted by the two secants

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)