What if you had a circle with two chords that share a common endpoint? How could you use the arc formed by those chords to determine the measure of the angle those chords make inside the circle? After completing this Concept, you'll be able to use the Inscribed Angle Theorem to solve problems like this one.
Watch This
Inscribed Angles in Circles CK-12
Guidance
An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle. The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc.
The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
Inscribed angles that intercept the same arc are congruent. This is called the Congruent Inscribed Angles Theorem and is shown below.
and intercept , so . Similarly, and intercept , so .
An angle intercepts a semicircle if and only if it is a right angle ( Semicircle Theorem ). Anytime a right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter and the diameter is the hypotenuse.
intercepts a semicircle, so . is a right angle, so is a semicircle.
Example A
Find and .
From the Inscribed Angle Theorem:
Example B
Find and .
The intercepted arc for both angles is . Therefore,
Example C
Find in .
is the center, so is a diameter. 's endpoints are on the diameter, so the central angle is .
Inscribed Angles in Circles CK-12
Guided Practice
1. Find and .
2. Fill in the blank: An inscribed angle is ____________ the measure of the intercepted arc.
3. Fill in the blank: A central angle is ________________ the measure of the intercepted arc.
Answers:
1. Use what you've learned about inscribed angles.
2. half
3. equal to
Practice
Fill in the blanks.
- An angle inscribed in a ________________ is .
- Two inscribed angles that intercept the same arc are _______________.
- The sides of an inscribed angle are ___________________.
- Draw inscribed angle in . Then draw central angle . How do the two angles relate?
Find the value of and/or in .
Solve for .
- Fill in the blanks of the Inscribed Angle Theorem proof.
Given : Inscribed and diameter
Prove :
Statement | Reason |
---|---|
1. Inscribed and diameter and |
1. |
2. | 2. |
3. | 3. All radii are congruent |
4. | 4. Definition of an isosceles triangle |
5. and | 5. |
6. and | 6. |
7. and | 7. |
8. | 8. Arc Addition Postulate |
9. | 9. |
10. | 10. Distributive PoE |
11. | 11. |
12. | 12. |