New York State Common Core Math Geometry, Module 5, Lesson 15
Worksheets for Geometry, Module 5, Lesson 15
- 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
- Shown below are circles with two intersecting secant chords.
Measure 𝑎, 𝑏, and 𝑐 in the two diagrams. Make a conjecture about the relationship between them.
𝒂 𝒃 𝒄
CONJECTURE about the relationship between 𝑎, 𝑏, and 𝑐:
- We will prove the following.
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.
We can interpret this statement in terms of the diagram below. Let 𝑏 and 𝑐 be the angle measures of the arcs
intercepted by ∠𝑆𝐴𝑄 and ∠𝑃𝐴𝑅. Then measure 𝑎 is the average of 𝑏 and 𝑐; that is, 𝑎 =
(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.)
Shown below are two circles with two secant chords intersecting outside the circle.
Measure 𝑎, 𝑏, and 𝑐. Make a conjecture about the relationship between them.
Conjecture about the relationship between 𝑎, 𝑏, and 𝑐:
Test your conjecture with another diagram.
Find 𝑥, 𝑦, and/or z
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
- 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
- SECANT ANGLE THEOREM—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.
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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