Secant Lines; Secant Lines That Meet Inside a Circle


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

Worksheets for Geometry

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

𝐷𝐡 is tangent to the circle as shown. a. Find the values of π‘Ž and 𝑏. b. Is 𝐢𝐡 a diameter of the circle? Explain.

Exercises 1–2

  1. In circle 𝑃, 𝑃𝑂 is a radius, and π‘šπ‘€π‘‚ = 142Β°. Find π‘šβˆ π‘€π‘‚π‘ƒ, and explain how you know.
  2. In the circle shown, π‘šπΆπΈ = 55Β°. Find π‘šβˆ π·πΈπΉ and π‘šπΈπΊ . Explain your answer.

Example

a. Find π‘₯. Justify your answer. b. Find π‘₯.

We can state the results of part (b) of this example as the following theorem: SECANT ANGLE THEOREMβ€”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 π‘₯ and οΏ½ 6. In the circle shown, 𝐡𝐢 is a diameter. Find π‘₯ and 𝑦. 7. In the circle shown, 𝐡𝐢 is a diameter. 𝐷𝐢: 𝐡𝐸 = 2: 1. Prove 𝑦 = 180 βˆ’ 3/2 π‘₯ using a two-column proof.

Lesson Summary

THEOREMS:

SECANT ANGLE THEOREMβ€”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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