The Inscribed Angle Alternate—A Tangent Angle

New York State Common Core Math Geometry, Module 5, Lesson 13

Worksheets for Geometry

Student Outcomes

• Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is tangent).
• Students solve a variety of missing angle problems using the inscribed angle theorem.

The Inscribed Angle Alternate—A Tangent Angle

Classwork

Opening Exercise

In circle 𝐴, 𝑚𝐵𝐷 = 56°, and 𝐵𝐶 is a diameter. Find the listed measure, and explain your answer.
a. 𝑚∠𝐵𝐷𝐶
b. 𝑚∠𝐵𝐶𝐷
c. 𝑚∠𝐷𝐵𝐶
d. 𝑚∠𝐵𝐹𝐺
e. 𝑚𝐵𝐶
f. 𝑚𝐷𝐶
g. Does ∠𝐵𝐺𝐷 measure 56°? Explain.
h. How do you think we could determine the measure of ∠𝐵𝐺𝐷?

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Exploratory Challenge

Examine the diagrams shown. Develop a conjecture about the relationship between 𝑎 and 𝑏.
Test your conjecture by using a protractor to measure 𝑎 and 𝑏.
Do your measurements confirm the relationship you found in your homework?
If needed, revise your conjecture about the relationship between 𝑎 and 𝑏:
Now, test your conjecture further using the circle below.

Now, we will prove your conjecture, which is stated below as a theorem.

THE TANGENT-SECANT THEOREM: Let 𝐴 be a point on a circle, let 𝐴𝐵 be a tangent ray to the circle, and let 𝐶 be a point on the circle such that 𝐴𝐶 is a secant to the circle. If 𝑎 = 𝑚∠𝐵𝐴𝐶 and 𝑏 is the angle measure of the arc intercepted by ∠𝐵𝐴𝐶, then 𝑎 = 1/2 𝑏.

Given circle 𝑂 with tangent 𝐴𝐵, prove what we have just discovered using what you know about the properties of a circle and tangent and secant lines.
a. Draw triangle 𝐴𝑂𝐶. What is the measure of ∠𝐴𝑂𝐶? Explain.
b. What is the measure of ∠𝑂𝐴𝐵? Explain.
c. Express the measure of the remaining two angles of triangle 𝐴𝑂𝐶 in terms of 𝑎 and explain.
d. What is the measure of ∠𝐴𝑂𝐶 in terms of 𝑎? Show how you got the answer.
e. Explain to your neighbor what we have just proven.

Exercises

Find 𝑥, 𝑦, 𝑎, 𝑏, and/or 𝑐.

Lesson Summary

THEOREMS:

• CONJECTURE: Let 𝐴 be a point on a circle, let 𝐴𝐵 be a tangent ray to the circle, and let 𝐶 be a point on the circle such that 𝐴𝐶 is a secant to the circle. If 𝑎 = 𝑚∠𝐵𝐴𝐶 and b is the angle measure of the arc intercepted by ∠𝐵𝐴𝐶, then 𝑎 = 1/2 𝑏.
• THE TANGENT-SECANT THEOREM: Let 𝐴 be a point on a circle, let 𝐴𝐵 be a tangent ray to the circle, and let 𝐶 be a point on the circle such that 𝐴𝐶 is a secant to the circle. If 𝑎 = 𝑚∠𝐵𝐴𝐶 and 𝑏 is the angle measure of the arc intercepted by ∠𝐵𝐴𝐶, then 𝑎 = 1/2 𝑏.
• Suppose 𝐴𝐵 is a chord of circle 𝐶, and 𝐴𝐷 is a tangent segment to the circle at point 𝐴. If 𝐸 is any point other than 𝐴 or 𝐵 in the arc of 𝐶 on the opposite side of 𝐴𝐵 from 𝐷, then 𝑚∠𝐵𝐸𝐴 = 𝑚∠𝐵𝐴𝐷.

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