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1-4 Measuring Angle 1-5 Exploring Angle Pairs p. 27 Objective: To find and compare the measures of angles To identify special angle pairs and use their relationships to find angle measure Angle: formed by two rays with the same endpoint B exterior The rays are the sides of the angle. The endpoint is the vertex of the angle Names:< ๐ช๐จ๐ฉ, < ๐ฉ๐จ๐ช, < ๐จ, < ๐ Vertex: ๐จ Sides: ๐จ๐ฉ, ๐จ๐ช A 1 interior C Problem 1 Naming Angles What are the two other names for < ๐? ๐ฑ ๐ฒ ๐ ๐ด Got it? 1a. What is two other names for < ๐ฒ๐ด๐ณ? b. Would it be correct to name any of the angles < ๐ด? Explain. ๐ ๐ณ Types of Angles acute angle ๐° ๐ < ๐ < ๐๐ right angle ๐° ๐ = ๐๐ obtuse angle ๐° ๐๐ < ๐ < ๐๐๐ straight angle ๐° ๐ = ๐๐๐ Congruent angles: angles with the same degree measure ๐จ ๐ฉ ๐<๐จ=๐<๐ฉ < ๐จ โ < ๐ฉ Problem 3 Using Congruent Angles Synchronized swimmers form angles with their bodies, as shown (p. 30). If ๐ < ๐ฎ๐ฏ๐ฑ = ๐๐, what is ๐ < ๐ฒ๐ณ๐ด? Angle Addition Postulate: If point ๐ฉ is in the interior of < ๐จ๐ถ๐ช then ๐ < ๐จ๐ถ๐ฉ + ๐ < ๐ฉ๐ถ๐ช = ๐ < ๐จ๐ถ๐ช ๐ฉ ๐จ ๐ถ ๐ช Problem 4 Using the Angle Addition Postulate If ๐ < ๐น๐ธ๐ป = ๐๐๐, what are the ๐ < ๐น๐ธ๐บ and ๐ < ๐ป๐ธ๐บ? ๐บ ๐น ๐๐ โ ๐๐ ๐๐ + ๐๐ Q ๐ป Adjacent angles: two coplanar angles with a common side, a common vertex, and no common interior points 1 2 Vertical angles: two angles formed by intersecting lines (sides are opposite rays) 3 1 2 <1 and <2 are vertical angles <3 and <4 are vertical angles โbowtiesโ 4 Vertical angle are congruent. Complementary angles: two angles whose measures have a sum of 90. Each angle is called a complement of the other Supplementary angles: two angles whose measures have a sum of 180 Each angle is called a supplement of the other p. 35 Problem 1 Identifying Angle Pairs Use the diagram at the right. Is the statement true? Explain a. < ๐ฉ๐ญ๐ซ and < ๐ช๐ญ๐ซ are adjacent angles. b. < ๐จ๐ญ๐ฉ and < ๐ฌ๐ญ๐ซ are vertical angles. c. < ๐จ๐ญ๐ฌ and < ๐ฉ๐ญ๐ช are complementary. ๐ฉ ๐จ ๐ญ ๐๐° ๐๐° ๐๐๐° ๐ฌ ๐ช ๐ซ Concept Summary p. 35 There are some relationships you can assume to be true from a diagram that has no marks or measures. You can conclude the following from an unmarked diagram. 1. Angles are adjacent. 2. Angles are adjacent and supplementary. 3. Angles are vertical angles. You cannot conclude the following from an unmarked diagram. 1. Angles or segments are congruent. 2. An angle is a right angle. 3. Angles are complementary. Problem 2 Making Conclusions From a Diagram What can you conclude from the information in the diagram? ๐ ๐ ๐ ๐ Got it? Can you make each conclusion from the information in the diagram? Explain. 2a. ๐ป๐พ โ ๐พ๐ฝ b. ๐ท๐พ โ ๐พ๐ธ c. < ๐ป๐พ๐ธ is a right angle d. ๐ป๐ฝ bisects ๐ท๐ธ ๐ Got it? Can you make each conclusion from the information in the diagram? Explain. 2a. ๐ป๐พ โ ๐พ๐ฝ b. ๐ท๐พ โ ๐พ๐ธ c. < ๐ป๐พ๐ธ is a right angle d. ๐ป๐ฝ bisects ๐ท๐ธ ๐ป ๐ธ ๐ท ๐พ ๐ฝ Linear Pair: a pair of adjacent angles whose noncommon sides are opposite rays Linear Pair Postulate: If two angles form a linear pair, then they are supplement <1 and <2 form a linear pair ๐ < ๐ + ๐ < ๐ = ๐๐๐ 1 2 Problem 3 Finding Missing Angle Measures < ๐ฒ๐ท๐ณ and < ๐ฑ๐ท๐ณ are a linear pair, ๐ < ๐ฒ๐ท๐ณ = ๐๐ + ๐๐, and ๐ < ๐ฑ๐ท๐ณ = ๐๐ + ๐๐. What are the measures of < ๐ฒ๐ท๐ณ and < ๐ฑ๐ท๐ณ? ๐ณ ๐ฒ ๐ท ๐ฑ Got it? 3a. How can you check your results in Problem 3? b. < ๐จ๐ซ๐ฉ and < ๐ฉ๐ซ๐ช are a linear pair, ๐ < ๐จ๐ซ๐ฉ = ๐๐ + ๐๐ and ๐ < ๐ฉ๐ซ๐ช = ๐๐ โ ๐. What are ๐ < ๐จ๐ซ๐ฉ and ๐ < ๐ฉ๐ซ๐ช? Angle bisector: a ray that divides an angle into two congruent angles 1 2 m ray m is an angle bisector < ๐ โ < ๐ Problem 4 Using an Angle Bisector to Find Angle Measure ๐จ๐ช bisects < ๐ซ๐จ๐ฉ. If the ๐ < ๐ซ๐จ๐ช = ๐๐, what is ๐ฆ < ๐ซ๐จ๐ฉ? ๐ซ ๐ช ๐๐° ๐จ ๐ฉ Got it? 4 ๐ฒ๐ด bisects < ๐ฑ๐ฒ๐ณ. If ๐ < ๐ฑ๐ฒ๐ณ = ๐๐, what is ๐ < ๐ฑ๐ฒ๐ด? HW p. 31 #6-16 even, 18-21 all p. 38 #8-30 even