Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Classwork Opening Exercise In a circle, a chord and a diameter are extended outside of the circle to meet at point . If = 46 , and = 32 , find . �𝐷𝐷𝐷𝐷�� 𝐴𝐴𝐴𝐴�� 𝐶𝐶 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 ⁰ 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 ⁰ 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

Let = , =

𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑦𝑦 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 𝑥𝑥 In , = Reason

∆𝐴𝐴𝐴𝐴𝐴𝐴 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = Reason

𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 46 + + + 90 = Reason

∴ 𝑥𝑥 𝑦𝑦 + =

𝑥𝑥 𝑦𝑦 In , = + 32 Reason

+∆𝐴𝐴𝐴𝐴𝐴𝐴+ 32𝑦𝑦= 𝑥𝑥 Reason

𝑥𝑥 𝑥𝑥 =

𝑥𝑥 = 𝑦𝑦 =

𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.39

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Exercises 1–4 Find the value in each figure below, and describe how you arrived at the answer.

1. Hint: Thales’𝑥𝑥 theorem 2.

3. 4.

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.40

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Lesson Summary:

THEOREMS:

• THE INSCRIBED ANGLE THEOREM: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle

• CONSEQUENCE OF INSCRIBED ANGLE THEOREM: Inscribed angles that intercept the same arc are equal in measure.

• If , , ’, and are four points with and ’ on the same side of line , and angles and are congruent, then , , ’, and all lie on the same circle. 𝐴𝐴 𝐵𝐵 𝐵𝐵 𝐶𝐶 𝐵𝐵 𝐵𝐵 ⃖𝐴𝐴��𝐴𝐴�⃗ ∠𝐴𝐴𝐴𝐴𝐴𝐴 ∠𝐴𝐴𝐴𝐴′𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐵𝐵 𝐶𝐶 Relevant Vocabulary

• CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.

• INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point.

• INTERCEPTED ARC: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc. An angle inscribed in a circle intercepts exactly one arc, in particular, the arc intercepted by a right angle is the semicircle in the interior of the angle.

Problem Set

In Problems 1–5, find the value . 1. 𝑥𝑥

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.41

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.42

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

6. If = , express in terms of .

𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹 𝑦𝑦 𝑥𝑥

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.43

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

7. a. Find the value .

𝑥𝑥

b. Suppose the = . Prove that = 3 .

𝑚𝑚∠𝐶𝐶 𝑎𝑎⁰ 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 𝑎𝑎⁰

8. In the figure below, three identical circles meet at , and , E respectively. = . , , and , , lie on straight lines. 𝐵𝐵 𝐹𝐹 𝐶𝐶 𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐹𝐹 𝐸𝐸 𝐷𝐷 Prove is a parallelogram.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.44

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY

PROOF:

Join and .

𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 = Reason: ______

𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 = ______= ______= ______= Reason: ______

𝑎𝑎 𝑑𝑑 ______= ______

Alternate angles are equal.

𝐴𝐴� ��𝐴𝐴�‖𝐹𝐹𝐹𝐹�� ______= ______

Corresponding angles are equal.

𝐴𝐴� ��𝐴𝐴�‖𝐵𝐵𝐵𝐵�� ______= ______

Corresponding angles are equal.

𝐵𝐵𝐵𝐵� ��‖�𝐶𝐶��𝐶𝐶�

𝐴𝐴� ��𝐴𝐴�‖𝐵𝐵𝐵𝐵��‖�𝐶𝐶��𝐶𝐶� is a parallelogram.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Date: 10/22/14 S.45