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More Lessons for Geometry

Common Core For Geometry

Worksheets for Geometry, Module 5, Lesson 7

Student Outcomes

- Define the angle measure of arcs, and understand that arcs of equal angle measure are similar.
- Restate and understand the inscribed angle theorem in terms of arcs: The measure of an inscribed angle is half the angle measure of its intercepted arc.
- Explain and understand that all circles are similar.

**The Angle Measure of an Arc**

Classwork

**Opening Exercise**

If the measure of β πΊπ΅πΉ is 17Β°, name three other
angles that have the same measure and explain why.

What is the measure of β πΊπ΄πΉ? Explain.

Can you find the measure of β π΅π΄π·? Explain.

**Example**

What if we started with an angle inscribed in the minor arc between π΄ and πΆ?

**Exercises**

- In circle π΄, ππ΅πΆ : ππΆπΈ : ππΈπ· : ππ·π΅ = 1: 2: 3: 4. Find the following angles of measure.

a. πβ π΅π΄πΆ

b. πβ π·π΄πΈ

c. ππ·π΅

d. ππΆπΈπ· - In circle π΅, π΄π΅ = πΆπ·. Find the following angles of measure.

a. ππΆπ·

b. ππΆπ΄π·

c. ππ΄πΆπ· - In circle π΄, π΅πΆ is a diameter and πβ π·π΄πΆ = 100Β°. If ππΈπΆ = 2ππ΅π· , find the following angles of measure.

a. πβ π΅π΄πΈ

b. ππΈπΆ

c. ππ·πΈπΆ - Given circle π΄ with πβ πΆπ΄π· = 37Β°, find the following angles of measure.

a. ππΆπ΅π·

b. πβ πΆπ΅π·

c. πβ πΆπΈπ·

**Lesson Summary**

**Theorems:**

- INSCRIBED ANGLE THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc.
- Two arcs (of possibly different circles) are similar if they have the same angle measure. Two arcs in the same or congruent circles are congruent if they have the same angle measure.
- All circles are similar

**Relevant Vocabulary**

- ARC: An arc is a portion of the circumference of a circle.
- MINOR AND MAJOR ARC: Let πΆ be a circle with center π, and let π΄ and π΅ be different points that lie on πΆ but are not the endpoints of the same diameter. The minor arc is the set containing π΄, π΅, and all points of πΆ that are in the interior of β π΄ππ΅. The major arc is the set containing π΄, π΅, and all points of πΆ that lie in the exterior of β π΄ππ΅.
- SEMICIRCLE: In a circle, let π΄ and π΅ be the endpoints of a diameter. A semicircle is the set containing π΄, π΅, and all points of the circle that lie in a given half-plane of the line determined by the diameter.
- 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.
- CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.
- INTERCEPTED ARC OF AN ANGLE: 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.

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