Secant Lines; Secant Lines That Meet Inside a Circle

New York State Common Core Math Geometry, Module 5, Lesson 14

Worksheets for Geometry, Module 5, Lesson 14

Student Outcomes

• Students understand that an angle whose vertex lies in the interior of a circle intersects the circle in two points and that the edges of the angles are contained within two secant lines of the circle.
• Students discover that the measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

Secant Lines; Secant Lines That Meet Inside a Circle

Classwork

Opening Exercise

𝐷𝐵 is tangent to the circle as shown. a. Find the values of 𝑎 and 𝑏. b. Is 𝐶𝐵 a diameter of the circle? Explain.

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Exercises 1–2

1. In circle 𝑃, 𝑃𝑂 is a radius, and 𝑚𝑀𝑂 = 142°. Find 𝑚∠𝑀𝑂𝑃, and explain how you know.
2. In the circle shown, 𝑚𝐶𝐸 = 55°. Find 𝑚∠𝐷𝐸𝐹 and 𝑚𝐸𝐺 . Explain your answer.

Example

a. Find 𝑥. Justify your answer. b. Find 𝑥.

We can state the results of part (b) of this example as the following theorem: SECANT ANGLE THEOREM—INTERIOR CASE: The measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

Exercises 3–7

In Exercises 3–5, find 𝑥 and � 6. In the circle shown, 𝐵𝐶 is a diameter. Find 𝑥 and 𝑦. 7. In the circle shown, 𝐵𝐶 is a diameter. 𝐷𝐶: 𝐵𝐸 = 2: 1. Prove 𝑦 = 180 − 3/2 𝑥 using a two-column proof.

Lesson Summary

THEOREMS:

SECANT ANGLE THEOREM—INTERIOR CASE: The measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

Relevant Vocabulary

• TANGENT TO A CIRCLE: A tangent line to a circle is a line in the same plane that intersects the circle in one and only one point. This point is called the point of tangency.
• TANGENT SEGMENT/RAY: A segment is a tangent segment to a circle if the line that contains it is tangent to the circle and one of the end points of the segment is a point of tangency. A ray is called a tangent ray to a circle if the line that contains it is tangent to the circle and the vertex of the ray is the point of tangency.
• SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two points.

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