Lesson 15: Secant Angle Theorem, Exterior Case

Home , Exterior angle theorem, Secant line

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

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

Lesson Notes The Opening Exercise reviews and solidifies the concept of secants intersecting inside of the circle and the relationships between the angles and the subtended arcs. Students then extend that knowledge in the Exploratory Challenge and Example. The Exploratory Challenge looks at a tangent and secant intersecting on the circle. The Example moves the point of intersection of two secant lines outside of the circle and continues to allow students to explore the angle/arc relationships.

Classwork Opening Exercise (10 minutes) This Opening Exercise reviews Lesson 14, secant lines that intersect inside circles. Students must have a firm understanding of this concept to extend this knowledge to secants intersecting outside the circle. Students need a protractor for this exercise. Have students work individually first. After their individual work, have students work with partners and compare answers. Use this as a way to informally assess student understanding.

Opening Exercise

1. Shown below are circles with two intersecting secant chords. Scaffolding: Post pictures of the different types of angle and arc relationships studied so far with the associated formulas to help students.

Measure 풂, 풃, and 풄 in the two diagrams. Make a conjecture about the relationship between them.

풂 풃 풄 ퟔퟎ° ퟖퟎ° ퟒퟎ° ퟏퟑퟎ° ퟏퟎퟎ° ퟏퟔퟎ°

CONJECTURE about the relationship between 풂, 풃, and 풄: 풃+풄 풂 = 풂 풃 풄 ퟐ . The measure is the average of and .

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 15 M5 GEOMETRY

2. 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 풃 and 풄 be the angle measures of the arcs 풃+풄 ∠푺푨푸 ∠푷푨푹 풂 풃 풄 풂 = intercepted by and . Then measure is the average of and ; that is, ퟐ .

a. Find as many pairs of congruent angles as you can in the diagram below. Express the measures of the angles in terms of 풃 and 풄 whenever possible.

ퟏ 풎∠푷푸푹 = 풎∠푷푺푹 = 풄 ퟐ ퟏ 풎∠푸푷푺 = 풎∠푸푹푺 = 풃 ퟐ

b. Which triangles in the diagram are similar? Explain how you know.

△ 푷푺푨 ∼ △ 푹푸푨. All angles in each pair have the same measure.

c. See if you can use one of the triangles to prove the secant angle theorem, interior case. (Hint: Use the exterior angle theorem.) MP.3 ퟏ 풂 = 풎∠푷푸푹 + 풎∠푸푹푺 풂 = (풃 + 풄) By the exterior angle theorem, . We can conclude ퟐ .

. Turn to your neighbor and summarize what we have learned so far in this exercise.

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Exploratory Challenge (10 minutes)

We have shown that the inscribed angle theorem can be extended to the case when one Scaffolding: of the angle’s rays is a tangent segment and the vertex is the point of tangency. The . For advanced learners, this Exploratory Challenge develops another theorem in the inscribed angle theorem’s family, Exploratory Challenge the secant angle theorem: exterior case. could be given as individual or pair work THEOREM. without leading questions.

SECANT ANGLE THEOREM—EXTERIOR CASE: The measure of an angle whose vertex lies in the . Use scaffolded questions exterior of the circle, and each of whose sides intersect the circle at two points, is equal to with a targeted small half the difference of the angle measures of its larger and smaller intercepted arcs. group.

. For example, look at the Exploratory Challenge table that you created. Do Shown below are two circles with two secant chords intersecting outside the circle. you see a pattern between the sum of 푏 and 푐 and the

value of 푎?

Figure 1 Figure 2

Measure 풂, 풃, and 풄. Make a conjecture about the relationship between them.

풂 풃 풄

Figure 1 ퟐퟑ° ퟐퟕ° ퟕퟑ°

Figure 2 ퟐퟓ° ퟐퟔ° ퟕퟔ°

Conjecture about the relationship between 풂, 풃, and 풄:

The value of 풂 is half the difference of the value of (풄 − 풃).

Test your conjecture with another diagram.

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Example (7 minutes)

In this example, we will rotate the secant lines one at a time until one and then both are tangent to the circle. This should be easy for students to see but can be shown with dynamic geometry software.

. Let’s go back to our circle with two secant lines intersecting in the exterior of the circle (show circle at right). . Remind me how I would find the measure of angle 퐶.  Half the difference between the longer intercepted arc and the shorter intercepted arc. 1  (푚퐷퐸̂ − 푚퐹퐺̂) 2 . Rotate one of the secant segments so that it becomes tangent to the circle (show circle at right). . Can we apply the same formula?  Answers will vary, but the answer is yes. . What is the longer intercepted arc? The shorter intercepted arc?  The longer arc is 퐷퐸̂ . The shorter arc is 퐷퐺̂ . . So, do you think we can apply the formula? Write the formula. 1  Yes. (푚퐷퐸̂ − 푚퐷퐺̂) 2 . Why is it not identical to the first formula?  Point 퐷 is an endpoint that separates the two arcs. . Now, rotate the other secant line so that it is tangent to the circle. (Show the circle to the right). . Does our formula still apply?  Answers will vary, but the answer is yes. . What is the longer intercepted arc? The shorter intercepted arc?  The longer arc is 퐷퐸̂ . The shorter arc is 퐸퐷̂ . . How can they be the same?  They are not. We need to add a point in between so that we can show they are two different arcs. . So, what is the longer intercepted arc? The shorter intercepted arc?  The longer arc is 퐷퐻퐸̂ . The shorter arc is 퐸퐷̂ .

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. So, do you think we can apply the formula? Write the formula. 1  Yes. (푚퐷퐻퐸̂ − 푚퐸퐷̂). 2 . Why is this formula different from the first two?  Points 퐷 and 퐸 are the endpoints that separate the two arcs. . Turn to your neighbor, and summarize what you have learned in this exercise.

Exercises (8 minutes) Have students work on the exercises individually and check their answers with a neighbor. Use this as an informal assessment, and clear up any misconceptions. Have students present problems to the class as a wrap-up.

Exercises

Find 풙, 풚, and/or 풛.

1. 2.

풙 = ퟐퟖ 풙 = ퟕퟐ

3. 4.

풙 = ퟑퟓ 풙 = ퟔퟔ, 풚 = ퟓퟕ, 풛 = ퟓퟕ

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Closing (5 minutes) Have students complete the summary table, and then share as a class to make sure students understand the concepts.

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.

The Inscribed Angle Theorem and its Family of Theorems

Relationship between Diagram How the two shapes overlap 풂, 풃, 풄, and 풅

Intersection is on the circle. ퟏ 풂 = 풃 ퟐ

(Inscribed Angle Theorem)

Intersection is on the circle. ퟏ 풂 = 풃 ퟐ

(Secant–Tangent)

Intersection is in the interior 풃 + 풄 풂 = of the circle. ퟐ

(Secant Angle Theorem—Interior)

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Intersection is exterior to the 풄 − 풃 풂 = circle. ퟐ

(Secant Angle Theorem—Exterior)

Intersection of two tangent 풄 − 풃 풂 = lines to a circle. ퟐ

(Two Tangent Lines)

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. . 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.

Relevant Vocabulary

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

Exit Ticket (5 minutes)

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Name Date

Lesson 15: Secant Angle Theorem, Exterior Case

Exit Ticket

1. Find 푥. Explain your answer.

2. Use the diagram to show that 푚퐷퐸̂ = 푦° + 푥° and 푚퐹퐺̂ = 푦° − 푥°. Justify your work.

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Exit Ticket Sample Solutions

1. Find 풙. Explain your answer.

풙 = ퟒퟎ. Major arc 풎푩푫̂ = ퟑퟔퟎ − ퟏퟒퟎ = ퟐퟐퟎ. ퟏ 풙 = (ퟐퟐퟎ − ퟏퟒퟎ) = ퟒퟎ ퟐ

2. Use the diagram to show that 풎푫푬̂ = 풚° + 풙° and 풎푭푮̂ = 풚° − 풙°. Justify your work. ퟏ 풙 = (풎푫푬̂ − 풎푭푮̂ ) ퟐ풙 = 풎푫푬̂ − 풎푭푮̂ ퟐ , or . Angle whose vertex lies exterior of circle is equal to half the difference of the angle measures of its larger and smaller intercepted arcs. ퟏ 풚 = (풎푫푬̂ + 풎푭푮̂ ) ퟐ풚 = 풎푫푬̂ + 풎푭푮̂ ퟐ , or . Angle whose vertex lies in a circle is equal to half the sum of the arcs intercepted by the angle and its vertical angle.

Adding the two equations gives ퟐ풙 + ퟐ풚 = ퟐ풎, or 풙 + 풚 = 풎푫푬̂ .

Subtracting the two equations gives ퟐ풚 − ퟐ풙 = ퟐ풎푭푮̂ , or 풚 − 풙 = 풎푭푮̂ .

Problem Set Sample Solutions

1. Find 풙. 2. Find 풎∠푫푭푬 and 풎∠푫푮푩.

풙 = ퟑퟐ 풎∠푫푭푬 = ퟔퟕ°, 풎∠푫푮푩 = ퟖퟖ°

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3. Find 풎∠푬푪푫, 풎∠푫푩푬, and 풎∠푫푬푩. 4. Find 풎∠푭푮푬 and 풎∠푭푯푬.

풎∠푬푪푫 = ퟔퟖ°, 풎∠푫푩푬 = ퟓퟔ°, 풎∠푫푬푩 = ퟔퟖ° 풎∠푭푮푬 = ퟐퟖ°, 풎∠푭푯푬 = ퟗퟔ°

5. Find 풙 and 풚.

풙 = ퟔퟒ°, 풚 = ퟑퟎ°

6. The radius of circle 푨 is ퟒ. 푫푪̅̅̅̅ and 푪푬̅̅̅̅ are tangent to the circle with 푫푪 = ퟏퟐ. Find 풎푫푬̂ and the area of quadrilateral 푫푨푬푪 rounded to the nearest hundredth. ퟒ 풎∠푪 = ퟐ (퐭퐚퐧−ퟏ ) ≈ ퟑퟔ. ퟖퟕ° ퟏퟐ Let 풎푫푬̂ be 풙. (ퟑퟔퟎ° − 풙) − 풙 = ퟑퟔ. ퟖퟕ° ퟐ 풙 = ퟏퟒퟑ. ퟏퟑ°

Therefore, 풎푫푬̂ = ퟏퟒퟑ. ퟏퟑ°, and the area is ퟒퟖ square

units .

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7. Find the measure of 푩푮̂ , 푭푩̂ , and 푮푭̂ . 8. Find the values of 풙 and 풚.

풎푩푮̂ = 풎푭푩̂ = ퟏퟏퟎ°, 풎푮푭̂ = ퟏퟒퟎ° 풙 = ퟏퟎퟓ, 풚 = ퟒퟎ

9. The radius of a circle is ퟔ. a. If the angle formed between two tangent lines to the circle is ퟔퟎ°, how long is the segment between the point of intersection of the tangent lines and the center of the circle? ퟏퟎ. ퟑퟗ

b. If the angle formed between the two tangent lines is ퟏퟐퟎ°, how long are each of the segments between the point of intersection of the tangent lines and the point of tangency? Round to the nearest hundredth. ퟔ ≈ ퟑ. ퟒퟔ √ퟑ

10. 푫푪̅̅̅̅ and 푬푪̅̅̅̅ are tangent to circle 푨. Prove 푩푫 = 푩푬.

Join 푨푫̅̅̅̅, 푨푬̅̅̅̅, 푩푫̅̅̅̅̅, and 푩푬̅̅̅̅.

푨푫 = 푨푬 Radii of same circle

푨푪 = 푨푪 Reflexive Property

풎∠푨푫푪 = 풎∠푨푬푪 = ퟗퟎ° Radii perpendicular to tangent lines at point of tangency

△ 푨푫푪 and △ 푨푬푪 are right triangles Definition of right triangle

△ 푨푫푪 ≅ △ 푨푬푪 HL

풎∠푪푨푬 = 풎∠푪푨푫 Corresponding angles of congruent triangles are equal in measure.

푩푫̂ ≅ 푩푬̂ Congruent angles intercept congruent arcs.

푩푫 = 푩푬 Congruent arcs intercept chords of equal measure.

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