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Common Core For Geometry

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**

π·π΅ is tangent to the circle as shown. a. Find the values of π and π. b. Is πΆπ΅ a diameter of the circle? Explain.

**Exercises 1β2**

- In circle π, ππ is a radius, and πππ = 142Β°. Find πβ πππ, and explain how you know.
- In the circle shown, ππΆπΈ = 55Β°. Find πβ π·πΈπΉ and ππΈπΊ . 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 THEOREMβINTERIOR 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 3β7**

In Exercises 3β5, find π₯ and οΏ½ 6. In the circle shown, π΅πΆ is a diameter. Find π₯ and π¦. 7. In the circle shown, π΅πΆ is a diameter. π·πΆ: π΅πΈ = 2: 1. Prove π¦ = 180 β 3/2 π₯ using a two-column proof.

**Lesson Summary**

**THEOREMS:**

SECANT ANGLE THEOREMβINTERIOR 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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