10 2 Measuring Angles and Arcs Reitz High

10. 2 Measuring Angles and Arcs Reitz High School

Targets: n Recognize major arcs, minor arcs, semicircles, and central angles and their measures n Find arc length

Central Angle n An angle with its vertex located at the center of a circle THE MEASURE OF A CENTRAL ANGLE IS THE SAME AS THE MEASURE OF ITS INTERCEPTED ARC.

Central Angle AOB.

Angles and Arcs n The sum of the measures of the central angles is 360°. m<1 + m<2 +m<3 +m<4 = 360

Minor Arc A minor arc is less then 180° and is labeled using the two endpoints. A major arc is greater than 180° but less than 360° and is labeled using the two endpoints and another point on the arc.

Minor Arc : Label with 2 endpoints: UV or VU

Major Arc: Label with 3 points: ACB

Semicircle n A semicircle measures 180° and is labeled using the two endpoints and another point on the arc.

Semicircle: Label with 3 letters: AKB, ACB, AHB

Angles and Arcs n Theorem 10. 1: In the same circle or circles, two arcs are iff their corresponding central angles are .

Postulate 10. 1: Arc Addition: n Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the arcs.

Arc Addition Sketch:

Example 1 a: ALGEBRA: Find Refer to. . Assume RV is a diameter.

Example 1 a: The sum of the measures of Substitution Simplify. Add 2 to each side. Divide each side by 26. Use the value of x to find Given Substitution Answer: 52

Example 1 b: ALGEBRA: Refer to Find . . Assume RV is a diameter.

Example 1 b: form a linear pair. Linear pairs are supplementary. Substitution Simplify. Subtract 140 from each side. Answer: 40

Your Turn: Refer to . Assume AD and BE are diameters. a. Find m Answer: 65 b. Find m Answer: 40

Example 2 a: In Find bisects. and

Example 2 a: is a minor arc, so is a semicircle. is a right angle. Arc Addition Postulate Substitution Subtract 90 from each side. Answer: 90

Example 2 b: In Find bisects. and

Example 2 b: since bisects . is a semicircle. Arc Addition Postulate Subtract 46 from each side. Answer: 67

Example 2 c: In Find bisects. and

Example 2 c: Vertical angles are congruent. Substitution. Subtract 46 from each side. Substitution. Subtract 44 from each side. Answer: 316

Your Turn: In and bisects a. Answer: 54 b. Answer: 72 c. Answer: 234 are diameters, Find each measure. and

Arc Length n

Example 3: In and . Find the length of .

Example 3: degree measure of arc degree measure of whole circle Answer: The length of arc length circumference is units or about 3. 14 units.

Your Turn: In and . Find the length of Answer: units or about 49. 48 units .