# Inscribed Angle Theorem and Its Applications

### New York State Common Core Math Geometry, Module 5, Lesson 5

Worksheets for Geometry, Module 5, Lesson 5

Student Outcomes

• Prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle.
• Recognize and use different cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure.

Inscribed Angle Theorem and Its Applications

Classwork

Opening Exercise

a. 𝐴 and 𝐶 are points on a circle with center 𝑂.
i. Draw a point 𝐵 on the circle so that 𝐴𝐵 is a diameter.
Then draw the angle 𝐴𝐵𝐶.
ii. What angle in your diagram is an inscribed angle?
iii. What angle in your diagram is a central angle?
iv. What is the intercepted arc of ∠𝐴𝐵𝐶?
v. What is the intercepted arc of ∠𝐴𝑂𝐶?
b. The measure of the inscribed angle 𝐴𝐶𝐷 is 𝑥, and the measure of the central angle 𝐶𝐴𝐵 is 𝑦. Find 𝑚∠𝐶𝐴𝐵 in terms of 𝑥

Example 1

𝐴 and 𝐶 are points on a circle with center 𝑂.
a. What is the intercepted arc of ∠𝐶𝑂𝐴? Color it red.
b. Draw triangle 𝐴𝑂𝐶. What type of triangle is it? Why?
c. What can you conclude about 𝑚∠𝑂𝐶𝐴 and 𝑚∠𝑂𝐴𝐶? Why?
d. Draw a point 𝐵 on the circle so that 𝑂 is in the interior of the inscribed angle 𝐴𝐵𝐶.
e. What is the intercepted arc of ∠𝐴𝐵𝐶? Color it green.
f. What do you notice about 𝐴𝐶 ?
g. Let the measure of ∠𝐴𝐵𝐶 be 𝑥 and the measure of ∠𝐴𝑂𝐶 be 𝑦. Can you prove that 𝑦 = 2𝑥? (Hint: Draw the diameter that contains point 𝐵.)
h. Does your conclusion support the inscribed angle theorem?
i. If we combine the Opening Exercise and this proof, have we finished proving the inscribed angle theorem?

Example 2

𝐴 and 𝐶 are points on a circle with center 𝑂.
a. Draw a point 𝐵 on the circle so that 𝑂 is in the exterior of the inscribed angle 𝐴𝐵𝐶.
b. What is the intercepted arc of ∠𝐴𝐵𝐶? Color it yellow.
c. Let the measure of ∠𝐴𝐵𝐶 be 𝑥 and the measure of ∠𝐴𝑂𝐶 be 𝑦. Can you prove that 𝑦 = 2𝑥? (Hint: Draw the diameter that contains point 𝐵.)
d. Does your conclusion support the inscribed angle theorem?
e. Have we finished proving the inscribed angle theorem?

Exercises

1. Find the measure of the angle with measure 𝑥. Diagrams are not drawn to scale.
2. Toby says △ 𝐵𝐸𝐴 is a right triangle because 𝑚∠𝐵𝐸𝐴 = 90°. Is he correct? Justify your answer.
3. Let’s look at relationships between inscribed angles.
a. Examine the inscribed polygon below. Express 𝑥 in terms of 𝑦 and 𝑦 in terms of 𝑥. Are the opposite angles in any quadrilateral inscribed in a circle supplementary? Explain.
b. Examine the diagram below. How many angles have the same measure, and what are their measures in terms of 𝑥˚?
4. Find the measures of the labeled angles

Lesson Summary

Theorems:

• THE INSCRIBED ANGLE THEOREM: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle.
• CONSEQUENCE OF INSCRIBED ANGLE THEOREM: Inscribed angles that intercept the same arc are equal in measure

Relevant Vocabulary

• INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point.
• INTERCEPTED ARC: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc. An angle inscribed in a circle intercepts exactly one arc, in particular, the arc intercepted by an inscribed right angle is the semicircle in the interior of the angle.

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