What if you were given a circle with either a chord and a tangent or two chords that meet at a common point? How could you use the measure of the arc(s) formed by those circle parts to find the measure of the angles they make on or inside the circle? After completing this Concept, you'll be able to apply the Chord/Tangent Angle Theorem and the Intersecting Chords Angle Theorem to solve problems like this one.
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Angles On and Inside a Circle CK12
Guidance
When we say an angle is on a circle, we mean the vertex is on the edge of the circle. One type of angle on a circle is the inscribed angle (see Inscribed Angles in Circles ). Another type of angle on a circle is one formed by a tangent and a chord.
Chord/Tangent Angle Theorem: The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.
If two angles, with their vertices on the circle, intercept the same arc then the angles are congruent.
An angle is inside a circle when the vertex lies anywhere inside the circle.
Intersecting Chords Angle Theorem: The measure of the angle formed by two chords that intersect inside a circle is the average of the measures of the intercepted arcs.
Example A
Find .
Use the Chord/Tangent Angle Theorem. .
Example B
Find and .
Example C
Find .
Use the Chord/Tangent Angle Theorem. .
Angles On and Inside a Circle CK12
Guided Practice
Find .
1.
2.
3.
Answers:
Use the Intersecting Chords Angle Theorem to write an equation.
1.
2.
3. is supplementary to the angle that is the average of the given intercepted arcs. We call this supplementary angle .
Practice
Find the value of the missing variable(s).
Solve for .
 Fill in the blanks of the proof for the Intersecting Chords Angle Theorem
Given : Intersecting chords and .
Prove :
Statement  Reason 

1. Intersecting chords and .  1. 
2. Draw

2. Construction 
3.  3. 
4.  4. 
5.  5. 
Fill in the blanks.
 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.