Geometry Chapter 10 10-2: Find Arc Measures

Find Arc Measures Objective: Students will be able to use angle measures to find measures for arcs of circles. Agenda Central Angle Arcs Measures of Arcs Congruent Arcs

Angle in the Circle Central Angle: An angle with its vertex at the center of the circle, created by two radii. <𝑨𝑪𝑩 is a central angle of ʘ 𝑪. 𝑪 𝑨 𝑩

Angle in the Circle Arc: A portion of the circle connecting two points from the circle. 𝑪 𝑨 𝑩 𝑨𝑩 is an arc of ʘ 𝑪

Types of Arcs Minor Arc: The shortest arc connecting two points.

Arcs by Image Minor Arc Notation: 𝑫𝑬

Types of Arcs Minor Arc: The shortest arc connecting two points. Semicircle: An arc that connects two points on opposite sides of the circle (i.e. the points of the diameter).

Arcs by Image Minor Arc Semicircle Notation: 𝑫𝑬 Notation: 𝑭𝑮𝑯

Types of Arcs Minor Arc: The shortest arc connecting two points. Semicircle: An arc that connects two points on opposite sides of the circle (i.e. the points of the diameter). Major Arc: The longest arc connecting two points.

Arcs by Image Minor Arc Semicircle Major Arc 𝑫𝑬 𝑭𝑮𝑯 𝑱𝑲𝑳 Notation:

Example 1 𝑨 𝑩 𝑶 𝑪 𝑭 𝑫 a.) 𝑨𝑩𝑫 b.) 𝑨𝑪 c.) 𝑨𝑫𝑩 d.) 𝑨𝑭𝑪 Identify the type of arc based off the picture and the notation. a.) 𝑨𝑩𝑫 b.) 𝑨𝑪 c.) 𝑨𝑫𝑩 d.) 𝑨𝑭𝑪 𝑨 𝑩 𝑶 𝑫 𝑪 𝑭

Example 1 𝑨 𝑩 𝑶 𝑪 𝑭 𝑫 a.) 𝑨𝑩𝑫 Semicircle b.) 𝑨𝑪 Minor Arc Identify the type of arc based off the picture and the notation. a.) 𝑨𝑩𝑫 Semicircle b.) 𝑨𝑪 Minor Arc c.) 𝑨𝑫𝑩 Major Arc d.) 𝑨𝑭𝑪 Major Arc 𝑨 𝑩 𝑶 𝑫 𝑪 𝑭

Measures of the Arcs The measure of a minor arc is equal to the measure of the central angle. The expression “𝒎 𝑫𝑬” is read as “the measure of arc 𝐷𝐸” Example 𝒎 𝑫𝑬=𝟕𝟎° Rule: 𝑴𝒊𝒏𝒐𝒓 𝑨𝒓𝒄 𝑴𝒆𝒂𝒔𝒖𝒓𝒆 = 𝑪𝒆𝒏𝒕𝒓𝒂𝒍 𝑨𝒏𝒈𝒍𝒆 𝑴𝒆𝒂𝒔𝒖𝒓𝒆 70°

Measures of the Arcs A semicircle always has a measure of 180°. Example 180° 𝒎 𝑭𝑮𝑯=𝟏𝟖𝟎° Rule: 𝑺𝒆𝒎𝒊𝒄𝒊𝒓𝒄𝒍𝒆 𝑴𝒆𝒂𝒔𝒖𝒓𝒆=𝟏𝟖𝟎°

Measures of the Arcs Example 300° 𝒎 𝑱𝑲𝑳=𝟑𝟎𝟎° Rule: As a circle has a measure of 360°, then we can take the measure of a major arc by taking the difference between 360° and the measure of the related minor arc. Example 300° 𝒎 𝑱𝑲𝑳=𝟑𝟎𝟎° Rule: 𝑴𝒂𝒋𝒐𝒓 𝑨𝒓𝒄 𝑴𝒆𝒂𝒔𝒖𝒓𝒆=𝟑𝟔𝟎 −𝑴𝒊𝒏𝒐𝒓 𝑨𝒓𝒄 𝑴𝒆𝒂𝒔𝒖𝒓𝒆

Example 2 Find the measure of each arc listed from ʘ 𝑷. a.) 𝑹𝑺 𝑹 𝑷 𝑺 𝑻 𝟏𝟏𝟎° b.) 𝑹𝑻𝑺 c.) 𝑹𝑺𝑻 d.) 𝑻𝑺

Example 2 Find the measure of each arc listed from ʘ 𝑷. a.) 𝑹𝑺 𝑹 𝑻 𝑅𝑆 is a minor arc, thus 𝒎 𝑹𝑺=𝒎<𝑹𝑷𝑺=𝟏𝟏𝟎° 𝟏𝟏𝟎°

Example 2 Find the measure of each arc listed from ʘ 𝑷. b.) 𝑹𝑻𝑺 𝑹 𝑅𝑇𝑆 is a major arc, thus 𝒎 𝑹𝑻𝑺=𝟑𝟔𝟎°−𝟏𝟏𝟎=𝟐𝟓𝟎° 𝟏𝟏𝟎°

Example 2 Find the measure of each arc listed from ʘ 𝑷. c.) 𝑹𝑺𝑻 𝑹 𝑅𝑆𝑇 is a semicircle, thus 𝒎 𝑹𝑺𝑻=𝟏𝟖𝟎° 𝟏𝟏𝟎°

Example 2 Find the measure of each arc listed from ʘ 𝑷. d.) 𝑻𝑺 𝑹 𝑇𝑆 is a minor arc, thus 𝒎 𝑻𝑺=𝒎<𝑻𝑷𝑺=𝟕𝟎° 𝟏𝟏𝟎° 𝟕𝟎°

Adding Arcs 𝒎 𝑨𝑩𝑪=𝒎 𝑨𝑩+𝒎 𝑩𝑪 Postulate 23 – Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measure of those two arcs. 𝑪 𝑨 𝑩 𝒎 𝑨𝑩𝑪=𝒎 𝑨𝑩+𝒎 𝑩𝑪

Example 3 Identify the given arcs as major arc, minor arc, or semicircle, and find the measure of the arc. 𝑸 𝑹 𝑺 𝟏𝟐𝟎° 𝑻 𝟖𝟎° 𝟔𝟎° a.) 𝑻𝑸 b.) 𝑸𝑻𝑺 d.) 𝑸𝑹𝑻 c.) 𝑻𝑺𝑹

Example 3 Identify the given arcs as major arc, minor arc, or semicircle, and find the measure of the arc. 𝑸 𝑹 𝑺 𝟏𝟐𝟎° 𝑻 𝟖𝟎° 𝟔𝟎° a.) 𝑻𝑸 𝑇𝑄 is a minor arc, thus 𝒎 𝑻𝑸=𝟏𝟐𝟎°

Example 3 Identify the given arcs as major arc, minor arc, or semicircle, and find the measure of the arc. 𝑸 𝑹 𝑺 𝟏𝟐𝟎° 𝑻 𝟖𝟎° 𝟔𝟎° b.) 𝑸𝑻𝑺 𝑄𝑇𝑆 is a major arc, thus 𝒎 𝑸𝑻𝑺=𝒎 𝑸𝑻+𝒎 𝑻𝑺 𝟏𝟐𝟎°+𝟖𝟎°=𝟐𝟎𝟎°

Example 3 Identify the given arcs as major arc, minor arc, or semicircle, and find the measure of the arc. c.) 𝑻𝑺𝑹 𝑸 𝑹 𝑺 𝟏𝟐𝟎° 𝑻 𝟖𝟎° 𝟔𝟎° 𝑇𝑆𝑅 is a semicircle, thus 𝒎 𝑻𝑺𝑹=𝟏𝟖𝟎°

Example 3 Identify the given arcs as major arc, minor arc, or semicircle, and find the measure of the arc. 𝑸 𝑹 𝑺 𝟏𝟐𝟎° 𝑻 𝟖𝟎° 𝟔𝟎° 𝟏𝟎𝟎° d.) 𝑸𝑹𝑻 𝑄𝑅𝑇 is a major arc, thus 𝒎 𝑸𝑹𝑻=𝒎 𝑸𝑹+𝒎 𝑹𝑺+𝒎 𝑻𝑺 𝟔𝟎°+𝟏𝟎𝟎°+𝟖𝟎°=𝟐𝟒𝟎°

Congruent Arcs Circles are congruent if they have the same radius. Two arcs are congruent arcs if they have the same measure and they are arcs of the same circle, or if they are arcs of congruent circles.

Example 4 Determine if the red arcs are congruent. Explain why or why not. a.) 𝑫 𝑬 𝑭 𝑪 𝟖𝟎°

Example 4 Determine if the red arcs are congruent. Explain why or why not. a.) 𝑫 𝑬 𝑭 𝑪 𝟖𝟎° Yes; 𝑪𝑫 and 𝑬𝑭 are in the same circle, and 𝒎 𝑪𝑫=𝒎 𝑬𝑭

Example 4 Determine if the red arcs are congruent. Explain why or why not. b.) 𝑻 𝑹 𝑼 𝑺

Example 4 Determine if the red arcs are congruent. Explain why or why not. b.) 𝑻 No; 𝒎 𝑹𝑺=𝒎 𝑻𝑼, but 𝑹𝑺 and 𝑻𝑼 are not in the same circle, nor are they in congruent circles. 𝑹 𝑼 𝑺

Example 4 Determine if the red arcs are congruent. Explain why or why not. c.) 𝑿 𝑽 𝟗𝟓° 𝒁 𝒀 𝟗𝟓°

Example 4 Determine if the red arcs are congruent. Explain why or why not. c.) 𝑿 𝑽 𝟗𝟓° 𝒁 𝒀 𝟗𝟓° Yes; 𝑽𝑿 and 𝒀𝒁 are in congruent circles, and 𝒎 𝑽𝑿=𝒎 𝒀𝒁.