Download Angle Theorems (part 1)

Aim: What are the theorems related to angles? (Day 1)
A
B
C
Do Now:
Given: EOD  BOA
OC bisects DOB
Prove: OC bisects AOE
Statement
D
E
O
Reason
1) EOD  BOA
1) Given
2) OC bisects DOB
2) Given
3) COD  BOC
3) Def. Angle Bisector
4) COD  EOD  BOC  BOA 4) Addition Post. (1,3)
5)COE  COD  EOD, COA  BOC  BOA 5) Partition Post.
6) COE  COA
6) Substitution Post.
1
Def.
Angle
Bisector
7) OC bisects AOE Geometry Lesson: Angle Theorems
7)
(Day 1)
Theorem #1: All right angles are congruent.
1
2
Given that 1 and 2 are right angles, what
statements can we make?
m1  90, m2  90
m1  m2
1  2
Def. Right Angles
Transitive Post.
Def. congruent angles
Geometry Lesson: Angle Theorems
(Day 1)
2
Def:
Def:Complementary Angles are two angles whose
measures add up to 90 degrees.
A
1
O
If AOC is a right angle, what can we say
about the sum m1  m2 ? m1  m2 =90
2
C
1 is "complementary" to 2. 2 is "complementary" to 1.
1 is "the complement" of 2. 2 is "the complement" of 1.
Geometry Lesson: Angle Theorems
(Day 1)
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Ex: Complementary Angles
In each case, A and B are complementary.
Determine mB for each:
1)
B
20º
3)
70º
90-x
B
x
A
A
2)
B
50º
40º
A
4) mA  3x  8
mB  90
_________
 (3x  8)
mB  98  3x
Geometry Lesson: Angle Theorems
(Day 1)
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Ex: Complementary Angles Proof A
Given: 1 is complementary to 2
3 is complementary to 2
Prove: m1  m3
Statement
1) 1 compl. 2
2) 3 compl. 2
3) m1  m2  90
4) m3  m2  90
5) m1  m2  m3  m2
6) m2  m2
7) m1  m3
1)
2)
3)
4)
5)
6)
7)
C
D
1 2
3
B
P
Reason
Given
Given
Def. comp. angles
Def. comp. angles
Substitution Postulate
Reflexive Postulate
Subtraction Postulate
Geometry Lesson: Angle Theorems
(Day 1)
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Theorems#2:
#2,Complements
3:
of the same angle, or congruent
Theorem
angles, are congruent.
2 is compl. 1
1
3 is compl. 1
What can we say about angles 2 and 3 ?
2  3
2
3
Theorem #3: If the union of two adjacent angles is a
right angle, the angles are complementary.
If ABC is a right angle, and
D
A
1
2
B
1  2  ABC , what can we
say about 1 and 2 ? 1 compl. 2
C
Geometry Lesson: Angle Theorems
(Day 1)
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Ex 1: Compl. Angles
Given: PA  PD PC  PB
Prove: 1  3
Statement
C
A
12
D
3
P
Reason
B
1) PA  PD PC  PB 1) Given
2) APD, CPB are rt.  ' s 2) Def. perpendicular lines
3) 1  2  APD
3) Partition Postulate
2  3  CPB
4) 1 compl. 2
3 compl. 2
5) 1  3
4) If the union of two adjacent angles
is a right angle, the angles are compl.
5)Complements of the same angle,
or congruent angles, are congruent.
Geometry Lesson: Angle Theorems
(Day 1)
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Ex 2,3,4: Compl. Angles:
2) Given: PB  AD, QC  AD
m1  m3
Prove: m2  m4
P•
•Q
4
2
3
1
•
A B
C
3) Given: PB  AD, 2 and 3 are complementary
Prove: m1  m3
•D
4) If A is the complement of B. If mA  5 x  6,
and mB  3x  4, what is:
a) x
b) mA
c) mB
Geometry Lesson: Angle Theorems
(Day 1)
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