Secant Angle Theorem, Exterior Case


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New York State Common Core Math Geometry, Module 5, Lesson 15

Worksheets for Geometry, Module 5, Lesson 15

Student Outcomes

  • Students find the measures of angle/arcs and chords in figures that include two secant lines meeting outside a circle, where the measures must be inferred from other data.

Secant Angle Theorem, Exterior Case

Classwork

Opening Exercise

  1. Shown below are circles with two intersecting secant chords. Measure π‘Ž, 𝑏, and 𝑐 in the two diagrams. Make a conjecture about the relationship between them. 𝒂 𝒃 𝒄 CONJECTURE about the relationship between π‘Ž, 𝑏, and 𝑐:
  2. We will prove the following. 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.
    We can interpret this statement in terms of the diagram below. Let 𝑏 and 𝑐 be the angle measures of the arcs intercepted by βˆ π‘†π΄π‘„ and βˆ π‘ƒπ΄π‘…. Then measure π‘Ž is the average of 𝑏 and 𝑐; that is, π‘Ž = (𝑏 + 𝑐)/2.
    a. Find as many pairs of congruent angles as you can in the diagram below. Express the measures of the angles in terms of 𝑏 and 𝑐 whenever possible.
    b. Which triangles in the diagram are similar? Explain how you know.
    c. See if you can use one of the triangles to prove the secant angle theorem, interior case. (Hint: Use the exterior angle theorem.)

Exploratory Challenge

Shown below are two circles with two secant chords intersecting outside the circle.
Measure π‘Ž, 𝑏, and 𝑐. Make a conjecture about the relationship between them.
Conjecture about the relationship between π‘Ž, 𝑏, and 𝑐:
Test your conjecture with another diagram.

Exercises

Find π‘₯, 𝑦, and/or z

Closing Exercise

We have just developed proofs for an entire family of theorems. Each theorem in this family deals with two shapes and how they overlap. The two shapes are two intersecting lines and a circle.
In this exercise, you will summarize the different cases

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.
  • SECANT ANGLE THEOREMβ€”EXTERIOR CASE: The measure of an angle whose vertex lies in the exterior of the circle, and each of whose sides intersect the circle in two points, is equal to half the difference of the angle measures of its larger and smaller intercepted arcs.

Relevant Vocabulary

SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two point




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