Download Document

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:< 𝑪𝑨𝑩, < 𝑩𝑨𝑪, < 𝑨, < 𝟏
Vertex: 𝑨
Sides: 𝑨𝑩, 𝑨𝑪
A
1
interior
C
Problem 1 Naming Angles
What are the two other names for < 𝟏?
𝑱
𝑲
𝟏
𝑴
Got it?
1a. What is two other names for < 𝑲𝑴𝑳?
b. Would it be correct to name any of the angles < 𝑴? Explain.
𝟐
𝑳
Types of Angles
acute angle
𝒙°
𝟎 < 𝒙 < 𝟗𝟎
right angle
𝒙°
𝒙 = 𝟗𝟎
obtuse angle
𝒙°
𝟗𝟎 < 𝒙 < 𝟏𝟖𝟎
straight angle
𝒙°
𝒙 = 𝟏𝟖𝟎
Congruent angles: angles with the same degree measure
𝑨
𝑩
𝒎<𝑨=𝒎<𝑩
< 𝑨 ≅< 𝑩
Problem 3 Using Congruent Angles
Synchronized swimmers form angles with their bodies, as shown (p. 30). If 𝒎 < 𝑮𝑯𝑱 = 𝟗𝟎,
what is 𝒎 < 𝑲𝑳𝑴?
Angle Addition Postulate: If point 𝑩 is in the interior of < 𝑨𝑶𝑪 then
𝒎 < 𝑨𝑶𝑩 + 𝒎 < 𝑩𝑶𝑪 = 𝒎 < 𝑨𝑶𝑪
𝑩
𝑨
𝑶
𝑪
Problem 4 Using the Angle Addition Postulate
If 𝒎 < 𝑹𝑸𝑻 = 𝟏𝟓𝟓, what are the 𝒎 < 𝑹𝑸𝑺 and 𝒎 < 𝑻𝑸𝑺?
𝑺
𝑹
𝟒𝒙 − 𝟐𝟎
𝟑𝒙 + 𝟏𝟒
Q
𝑻
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
“bowties”
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. < 𝑩𝑭𝑫 and < 𝑪𝑭𝑫 are adjacent angles.
b. < 𝑨𝑭𝑩 and < 𝑬𝑭𝑫 are vertical angles.
c. < 𝑨𝑭𝑬 and < 𝑩𝑭𝑪 are complementary.
𝑩
𝑨
𝑭 𝟐𝟖°
𝟔𝟐°
𝟏𝟏𝟖°
𝑬
𝑪
𝑫
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?
𝟐
𝟏
𝟑
𝟓
Got it?
Can you make each conclusion from the information in the diagram? Explain.
2a. 𝑻𝑾 ≅ 𝑾𝑽
b. 𝑷𝑾 ≅ 𝑾𝑸
c. < 𝑻𝑾𝑸 is a right angle
d. 𝑻𝑽 bisects 𝑷𝑸
𝟒
Got it?
Can you make each conclusion from the information in the diagram? Explain.
2a. 𝑻𝑾 ≅ 𝑾𝑽
b. 𝑷𝑾 ≅ 𝑾𝑸
c. < 𝑻𝑾𝑸 is a right angle
d. 𝑻𝑽 bisects 𝑷𝑸
𝑻
𝑸
𝑷
𝑾
𝑽
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
𝒎 < 𝟏 + 𝒎 < 𝟐 = 𝟏𝟖𝟎
1
2
Problem 3 Finding Missing Angle Measures
< 𝑲𝑷𝑳 and < 𝑱𝑷𝑳 are a linear pair, 𝒎 < 𝑲𝑷𝑳 = 𝟐𝒙 + 𝟐𝟒, and 𝒎 < 𝑱𝑷𝑳 = 𝟒𝒙 + 𝟑𝟔.
What are the measures of < 𝑲𝑷𝑳 and < 𝑱𝑷𝑳?
𝑳
𝑲
𝑷
𝑱
Got it?
3a. How can you check your results in Problem 3?
b. < 𝑨𝑫𝑩 and < 𝑩𝑫𝑪 are a linear pair, 𝒎 < 𝑨𝑫𝑩 = 𝟑𝒙 + 𝟏𝟒 and 𝒎 < 𝑩𝑫𝑪 = 𝟓𝒙 − 𝟐.
What are 𝒎 < 𝑨𝑫𝑩 and 𝒎 < 𝑩𝑫𝑪?
Angle bisector: a ray that divides an angle into two congruent angles
1
2
m
ray m is an angle bisector
< 𝟏 ≅< 𝟐
Problem 4 Using an Angle Bisector to Find Angle Measure
𝑨𝑪 bisects < 𝑫𝑨𝑩. If the 𝒎 < 𝑫𝑨𝑪 = 𝟓𝟖, what is 𝐦 < 𝑫𝑨𝑩?
𝑫
𝑪
𝟓𝟖°
𝑨
𝑩
Got it?
4 𝑲𝑴 bisects < 𝑱𝑲𝑳. If 𝒎 < 𝑱𝑲𝑳 = 𝟕𝟐, what is 𝒎 < 𝑱𝑲𝑴?
HW p. 31 #6-16 even, 18-21 all
p. 38 #8-30 even