Download Lesson 2.5 Angle Relationships notes

2.5 Angle Relationships
________________________ - Two adjacent angles whose distinct sides lie on the same line.
________________________ - Two nonadjacent angles formed by two intersecting lines.
__________________________ - A statement that can be expressed in “if-then” form.
IF a, THEN b.
 If an object is a triangle, then it is a polygon.
 If a number is even, then it is divisible by two.
________________________________________ - The statement formed by interchanging the
hypothesis (the “if” part) and conclusion (the “then” part) of a conditional statement.
IF b, THEN a.
 If an object is a polygon, then it is a triangle.
 If a number is divisible by two, then it is even.
_____________________________________ - The statement formed by negating both the
hypothesis (the “if” part) and the conclusion (the “then” part) of a conditional statement.
IF NOT a, THEN NOT b.
 If an object is not a triangle, then it is not a polygon.
 If a number is not even, then it is not divisible by two.
___________________________________ - The statement formed by interchanging and negating
both the hypothesis (the “if” part) and the conclusion (the “then” part) of a conditional statement.
(The inverse of the converse of a conditional statement.)
IF NOT b, THEN NOT a.
 If an object is not a polygon, then it is not a triangle.
 If a number is not divisible by two, then it is not even.
Geometry Lesson 2.5: Angle Relationships
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Investigation: Linear Pair Conjecture
Step 1 Draw ⃡𝑃𝑄 and place a point R between P and Q. Choose another point S not on ⃡𝑃𝑄 and draw
𝑅𝑆. You have just created a linear pair of angles. Place the “zero edge” of your protractor
along ⃡𝑃𝑄 . What do you notice about the sum of the measures of the pair of angles?
Step 2 Compare. Does everyone make the same observation? Complete the statement.
If two angles form a _____________, then the measures of the angles add up to _____.
D
mACD + mDCB = ______
A
B
C
Investigation: Vertical Angles Conjecture
Step 1 Draw two intersecting lines onto patty paper. Label the angles as shown in the box below.
Which angles are vertical angles?
Step 2: Fold the paper so that the vertical angles lie over each other. What do you notice about their
measures?
Step 3 Repeat this investigation with another pair of intersecting lines.
Step 4 Compare your results with the results of others. Complete the statement.
If two angles are __________________, then they are _____________.
1
4
2
1   3 and 2  4
3
Geometry Lesson 2.5: Angle Relationships
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Example 1: Write the converse, inverse, and contrapositive of each and tell whether each is true or
false.
a) Linear pair conjecture:
converse:
inverse:
contrapositive:
b) Vertical angle conjecture:
converse:
inverse:
contrapositive:
Example 2: Find x, y, and z.
135
z
x
y
Geometry Lesson 2.5: Angle Relationships
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Example 3: Find all the missing angles.
109
a
c
e
d
50
b
150
a = ______
b = ______
d = _____
c = ______
h
e = ______
f = ______
130
g
f
55
k
i
j
120
v
l
q
m
110
n
105
145
p
u
r
t
o
s
g = ______
h = ______
i = ______
p = ______
q = ______
r = ______
j = ______
k = ______
l = ______
s = ______
t = ______
u = ______
m = ______
n = ______
o = ______
v = ______
Geometry Lesson 2.5: Angle Relationships
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Example 4: Find the value of x in each.
20x - 40
5x + 20
6x + 10
11x - 15
22x - 34
5x + 25
15x - 30
8x - 20
4x + 20
7x + 50
Homework: pp. 122 – 125 => 1 – 10; 13 – 17; 19; 22 – 27
Geometry Lesson 2.5: Angle Relationships
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