Learning Objectives
 Find the measures of angles formed by chords, secants, and tangents.
Review Queue
 What is and ? How do you know?
 Find .
 Find .
Know What? The sun’s rays hit the Earth such that the tangent rays determine when daytime and night time are. If the arc that is exposed to sunlight is , what is the angle at which the sun’s rays hit the earth ?
Angle on a Circle
When an angle is on a circle, the vertex is on the edge of the circle. One type of angle on a circle is the inscribed angle, from the previous section. Another type of angle on a circle is one formed by a tangent and a chord.
Investigation 96: The Measure of an Angle formed by a Tangent and a Chord
Tools Needed: pencil, paper, ruler, compass, protractor
 Draw with chord and tangent line with point of tangency .
 Draw in central angle . Find and .
 Find . How does the measure of this arc relate to ?
Theorem 911: The measure of an angle formed by a chord and a tangent that intersect on the circle is half the measure of the intercepted arc.
We now know that there are two types of angles that are half the measure of the intercepted arc; an inscribed angle and an angle formed by a chord and a tangent.
Example 1: Find:
a)
b)
Solution: Use Theorem 911.
a)
b)
Example 2: Find and .
Solution:
From this example, we see that Theorem 98 is true for angles formed by a tangent and chord with the vertex on the circle. If two angles, with their vertices on the circle, intercept the same arc then the angles are congruent.
Angles inside a Circle
An angle is inside a circle when the vertex anywhere inside the circle, but not on the center.
Investigation 97: Find the Measure of an Angle inside a Circle
Tools Needed: pencil, paper, compass, ruler, protractor, colored pencils (optional)
 Draw with chord and . Label the point of intersection .
 Draw central angles and . Use colored pencils, if desired.
 Find and . Find and .
 Find .
 What do you notice?
Theorem 912: The measure of the angle formed by two chords that intersect inside a circle is the average of the measure of the intercepted arcs.
Example 3: Find .
a)
b)
c)
Solution: Use Theorem 912 to write an equation.
a)
b)
c) is supplementary to the angle that the average of the given intercepted arcs, .
Angles outside a Circle
An angle is outside a circle if the vertex of the angle is outside the circle and the sides are tangents or secants. The possibilities are: an angle formed by two tangents, an angle formed by a tangent and a secant, and an angle formed by two secants.
Investigation 98: Find the Measure of an Angle outside a Circle
Tools Needed: pencil, paper, ruler, compass, protractor, colored pencils (optional)
 Draw three circles and label the centers and . In draw two secant rays with the same endpoint. In , draw two tangent rays with the same endpoint. In , draw a tangent ray and a secant ray with the same endpoint. Label the points like the pictures below.
 Draw in all the central angles. Using a protractor, measure the central angles and find the measures of each intercepted arc.
 Find and .
 Find and . What do you notice?
Theorem 913: The measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs.
Example 4: Find the measure of .
a)
b)
c)
Solution: For all of the above problems we can use Theorem 913.
a)
b) is not the intercepted arc. The intercepted arc is .
c) Find the other intercepted arc,
Know What? Revisited From Theorem 913, we know .
Review Questions
 Questions 13 use the definitions of tangent and secant lines.
 Questions 47 use the definition and theorems learned in this section.
 Questions 825 are similar to Examples 14.
 Questions 26 and 27 are similar to Example 4, but also a challenge.
 Questions 28 and 29 are fillintheblank proofs of Theorems 912 and 913.
 Draw two secants that intersect:
 inside a circle.
 on a circle.
 outside a circle.
 Can two tangent lines intersect inside a circle? Why or why not?
 Draw a tangent and a secant that intersect:
 on a circle.
 outside a circle.
Fill in the blanks.
 If the vertex of an angle is on the _______________ of a circle, then its measure is _______________ to the intercepted arc.
 If the vertex of an angle is _______________ a circle, then its measure is the average of the __________________ arcs.
 If the vertex of an angle is ________ a circle, then its measure is ______________ the intercepted arc.
 If the vertex of an angle is ____________ a circle, then its measure is ___________ the difference of the intercepted arcs.
For questions 825, find the value of the missing variable(s).
Challenge Solve for .
 Fill in the blanks of the proof for Theorem 912. Given: Intersecting chords and . Prove:
Statement  Reason 

1. Intersecting chords and .  
2. Draw

Construction 
3.  
4.  
5. 
 Fill in the blanks of the proof for Theorem 913. Given: Secant rays and Prove:
Statement  Reason 

1. Intersecting secants and .  
2. Draw .

Construction 
3.  
5.  
6.  Subtraction PoE 
7.  Substitution 
8. 
Review Queue Answers
 because a tangent line and a radius drawn to the point of tangency are perpendicular.