The Inscribed Angle Alternate—A Tangent Angle
In circle 𝐴, 𝑚𝐵𝐷 = 56°, and 𝐵𝐶 is a diameter. Find the listed measure, and explain your answer.
g. Does ∠𝐵𝐺𝐷 measure 56°? Explain.
h. How do you think we could determine the measure of ∠𝐵𝐺𝐷?
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
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.
Find 𝑥, 𝑦, 𝑎, 𝑏, and/or 𝑐.
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