9-2 Study Guide and Intervention Measuring Angles and Arcs

NAME ______DATE ______PERIOD ______

Angles and Arcs A central angle is an angle whose vertex is 퐺퐹̂ is a minor arc. at the center of a circle and whose sides are radii. A central angle 퐶퐻퐺̂ is a major arc. separates a circle into two arcs, a major arc and a minor arc. ∠GEF is a central angle. Here are some properties of central angles and arcs. • The sum of the measures of the central angles of a circle with

no interior points in common is 360. m∠HEC + m∠CEF + m∠FEG + m∠GEH = 360 • The measure of a minor arc is less than 180 and equal to the measure of its central angle. m퐶퐹̂ = m∠CEF • The measure of a major arc is 360 minus the measure of the minor arc. m퐶퐺퐹̂ = 360 – m퐶퐹̂

• The measure of a semicircle is 180. • Two minor arcs are congruent if and only if their corresponding central angles are congruent. 퐶퐹̂ ≅ 퐹퐺̂ if and only if ∠CEF ≅ ∠FEG. • The measure of an arc formed by two adjacent arcs is the sum ̂ ̂ ̂ of the measures of the two arcs. (Arc Addition Postulate) m퐶퐹 + m퐹퐺 = m퐶퐺

Example: 푨푪̅̅̅̅ is a diameter of ⨀R. Find m푨푩̂ and m푨푪푩̂.

∠ARB is a central angle and m∠ARB = 42, so m퐴퐵̂ = 42. Thus m퐴퐶퐵̂ = 360 – 42 or 318.

Exercises Find the value of x.

1. 80 2. 150

̅̅̅̅̅ ̅̅̅̅ 푩푫 and 푨푪 are diameters of ⨀O. Identify each arc as a major arc, minor arc, or semicircle of the circle. Then find its measure.

3. m퐵퐴̂ minor arc; 44 4. m 퐵퐶̂ minor arc; 136 5. m퐶퐷̂ minor arc; 44 6. m 퐴퐶퐵̂ major arc; 316 7. m퐵퐶퐷̂ semicircle; 180 8. m 퐴퐷̂ minor arc; 136

Chapter 9 11 Glencoe Geometry NAME ______DATE ______PERIOD ______

9-2 Study Guide and Intervention (continued) Measuring Angles and Arcs

Arc Length An arc is part of a circle and its length is a part of the circumference of the circle. The length of arc ℓ can be found using the following equation: 푥 ℓ = ⋅ 2πr 360

Example: Find the length of 푨푩̂ . Round to the nearest hundredth. 푥 The length of arc 퐴퐵̂ can be found using the following equation: 퐴퐵̂ = · 2πr 360 푥 퐴퐵̂ = · 2πr Arc Length Equation 360 135 퐴퐵̂ = · 2π(8) Substitution 360 퐴퐵̂ ≈ 18.85 in. Use a calculator.

Exercises Use ⨀O to find the length of each arc. Round to the nearest hundredth.

1. 퐷퐸̂ if the radius is 2 meters 4.19 m

2. 퐷퐸퐴̂ if the diameter is 7 inches 12.83 in.

3. 퐵퐶̂ if BE = 24 feet 9.42 ft

4. 퐶퐵퐴̂ if DO = 3 millimeters 7.07 mm

Use ⨀P to find the length of each arc. Round to the nearest hundredth.

5. 푅푇̂ , if MT = 7 yards 3.05 yd

6. 푀푅̂ , if PR = 13 feet 29.50 ft

7. 푀푆푇̂, if MP = 2 inches 6.28 in.

8. 푀푅푆̂, if PS = 10 centimeters 40.14 cm

Chapter 9 12 Glencoe Geometry