10.2 Measuring Angles and Arcs Reitz High School.

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Presentation transcript:

10.2 Measuring Angles and Arcs Reitz High School

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

Central Angle 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 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 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 Theorem 10.1: In the same circle or  circles, two arcs are  iff their corresponding central angles are .

Postulate 10.1: Arc Addition: 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 1a: ALGEBRA: Refer to . Assume RV is a diameter. Find .

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

Example 1b: 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 b. Find m Answer: 65 Answer: 40

Example 2a: In bisects and Find .

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

Example 2b: In bisects and Find .

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

Example 2c: In bisects and Find .

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

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

Arc Length Another way to measure an arc is by its length. An arc is part of a circle, so its length is part of the circumference. Arc Length = 𝑚 360 2⫪r m= measure of central angle r= radius of circle

Example 3: In and . Find the length of .

Example 3: degree measure of arc degree measure of whole circle arc length circumference Answer: The length of is units or about 3.14 units.

Your Turn: In and . Find the length of . Answer: units or about 49.48 units