Download Geometry-Chapter 2 Notes

Geometry-Chapter 2 Notes
2.1 Perpendicularity
Vocabulary:
perpendicular symbol___________ parallel symbol__________
Definition #16 Lines, rays or segments that intersect at right angles
are perpendicular.
Examples and label drawings and write the notations for each.
a) draw line “a” perpendicular
b) draw right angle DEF and give
to line “b”.
the notation that the sides of the
angle are perpendicular
c) draw perpendicular rays JM
and ray GH that intersect at K
d) draw the angle in “b” again
with a different 90˚ notation.
Do not assume perpendicularity or parallelness from a diagram!!!!!!!!!
Draw a coordinate plane below. Label both axes, the quadrants, the
origin and graph one ordered pair in each quadrant. Check your
neighbors work and let them check yours.
Given: AB ⊥ BC
DC ⊥ BC
A
2
D
Conclusion: ∠ B ≅ ∠ C
Statements
1) AB ⊥ BC
2)___________________
3) DC ⊥ BC
4) ∠ C is_____________
5)____________________
B
C
Reasons
1)________________
2) If two segments are ⊥ , they
form right angles.
3)________________
4)________________
5)_______________________
* * * *
* * * * *
* *
Given that EH ⊥ HG
Name all the angles you can prove to be
right angles.
*
*
*
H
E
*
F
*
*
G
___________________________________________________
Given KJ ⊥ KM
∠ JKO is four times as large as
∠ MKO. Find m ∠ JKO
J
K
4x
x
O
M
*
Given EC || to x axis
RT || to the x axis
Find the area
of rectangle RECT
E ( -4,3)
Y
3
C ( 7,3)
?
X
R (-4,-2)
?
T ( 7,-2)
____________________________________________________
2.2 Complementary and Supplementary Angles
Def. #17 Complementary angles are two angles whose sum is 90º.
Each of the two angles is called a complement of the other.
Draw and label the following complementary angles:
a) angle A is complementary b) a triangle with 2 complementary angles
c) draw and label a right angle divided into 2 complimentary angles
with measures of 63º 40’ and 26º 20’
Def. #18 supplementary angles are two angles whose sum is 180º. Each
of the two angles is called the supplement of the other.
Draw and label the following supplementary angles
a) angle B is supplementary
b) a quadrilateral with atleast one
supplementary angle
4
c) a straight angle divided into 2 angles, one of which is supplementary.
One angles is 131º 27’ 48” and the other is 48º 32’ 12”
Given:
Prove :
TVK is a right angle
1 is comp. to
2
T
X
1
Statement
1) _____________________
∠ 1 is comp. to ∠ 2
2)
Given: Diagram as shown
Conclusion: ∠ 1 is supp to ∠ 2
Statements
1)_____________________
2)______ is a straight angle
3) ∠ 1 is supp to ∠ 2
2
V
K
Reasons
1) Given
2)_______________________
1
2
A
B
C
Reasons
1) Given
2) _________________________
3) ________________________
The measure of one of two complementary angles is three more than
twice the measure of the other. Find the measure of each.
5
The measure of the supplement of an angle is 60 less than 3 times the
measure of the complement of the angle. Find the measure of the
complement.
____________________________________________________
2.3 Drawing Conclusions:
Procedure for Drawing Conclusions
1)
2)
3)
4)
5)
Given: AB bisects ∠ CAD
Conclusion:___________________
6
C
B
D
A
Statement
1) ____________________
2)_____________________
Given:
A is a right angle
B is a right angle
Conclusion_________________
Statements
Reasons
1) Given
2)_________________________
A
D
B
C
Given: E is the midpoint of SG
Conclusion:___________________
Statements
Reasons
S
E
Reasons
G
Given: ∠ PRS is a right angle
Conclusion:________________
7
P
R
Reasons
Statements
S
____________________________________________________
2.4 Congruent supplements and complements
Theorem #4 If angles are supplements to the same angle, then they
are congruent.
Theorem #5 If angles are supplementary to congruent angles, then
they are congruent
Theorem #6 If angles are complementary to the same angle, they are
congruent.
Theorem #7 If angles are complementary to congruent angles, then
they are congruent.
H
Sample #3
G
Given: Diagram as shown
F
Hint: # angles
Prove: ∠ HFE ≅ ∠ GFJ
E
J
Statement
Reason
1) Diagram as shown
1)_____________
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
Given: KM ⊥ MO
PO ⊥ MO
∠ KMR ≅ ∠ POR
Prove: ∠ ROM ≅ ∠ RMO
1)
Statements
K
R
M
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
Reasons
P
8
O
____________________________________________________
2.5 Addition and Subtraction Properties
Theorem #8 If a segment is added to two congruent segments the
sums are congruent.(Addition Property)
Theorem #9 If an angle is added to two congruent angles, the sums are
congruent. ( Addition Property)
Theorem #10 If congruent segments are added to congruent segments,
the sums are congruent. (Addition Property)
Theorem #11 If congruent angles are added to congruent angles, the
sums are congruent. (Addition Property)
9
SUBTRACTION PROPERTY
Theorem 12-If a segment(or angle) is subtracted from congruent
segments(or angles), the differences are congruent. (Subtraction
Property)
Theorem 13- If congruent segments ( or angles) are subtracted from
congruent segments ( or angles), the differences are congruent.
( Subtraction Property)
Using the Addition and Subtraction Properties in Proofs
1)
2)
Given ∠ NOP ≅ ∠ NPO
∠ ROP ≅ ∠ RPO
Prove: ∠ NOR ≅ ∠ NPR
Statements
N
R
O
Reasons
P
Given: AB ≅ CD
Conclusion________________
E
C
A
Statements
1)
2)
1)
D
B
10
F
Reasons
2)
____________________________________________________
Given :
H
E
F
Conclusion: ∠ HEF ≅ ∠ GFE
Statements
Reasons
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
G
11
2.6 Multiplication and Division Properties
With these properties the “multiples” and “divisions” of angles or
segments must be congruent to each other. (* You must state their
congruence in your proof.)
Theorem # 14 : If segments (or angles) are congruent, their like
multiples are congruent. ( Multiplication Property)
Example
A
B
C
D
E
F
G
H
* In the segments above B, C, F, and G are trisection points.
If segment AB is congruent to segment EF and they are both are 3
units long. What can we now say about segments AD and EH ?
____________________________________________________
J
N
O
S
K
P
M
R
Example: Rays KO and PS are angle bisectors. If angle JKO ≅ angle
NPS and they are both 25º , what can we say about angle JKM and
angle NPR?___________________________________________
Given: MK ≅ FG
M
K
KG bisects MJ and FH
Prove: MJ ≅ FH
Statement
F
G
Reason
H
J
Using the Multiplication and Division Properties in Proofs
12
Theorem # 15 If segments (or angles) are congruent, their like
divisions are congruent ( Division Property)
Example A
B
C
D
E
F
G H
* In the segments above segments AD and EH are congruent.
Points B, C, F, G are trisection points.
Prove that segment AB is congruent to segment EF.
What can we now conclude that will help us prove it
?_______________
___________________________________________________
J
N
O
K
M
S
P
R
Example: Angle JKM ≅ angle NPR . Rays KO and PS are angle
bisectors. What can we conclude about angle JKO and angle SPR?
__________________________________________________________
13
Given: ∠ NOP ≅ ∠ RPO
PT bisects RPO
OS bisects NOP
∠ NSO is comp. to 1
∠ RTP is comp to 3
Prove: ∠ NSO ≅ ∠ RTP
Statements
N
T
S
1
O
R
3
P
Reason
2.7 Transitive and Substitution Properties
Theorem # 16 If angles (or segments) are congruent to the same angle
(or segment), they are congruent to each other. (Transitive Property)
Ex. If angle A ≅ angle B, angle A ≅ angle C, is angle B ≅ angle C ?
Write as a chain of reasoning:
Theorem #17 If angles ( or segments) are congruent to congruent
angles (or segments) , they are congruent to each other.
• Make up your own example for this
property._______________________
14
Substitution:
A) You have used substitution this year to solve systems of equations in
which you have both an x and y. You substituted a value for X to
eliminate one of the variables.
Given: angle A ≅ angle B , solve for angle A
B
(x + 10)˚
A
(2x -4)˚
When do you substitute in this problem?__________________________
B) You can also apply substitution with out using any variables.
If ∠ 1 is comp ∠ 2 , and ∠ 2 ≅ ∠ 3 , then ∠ ___ is comp ∠ _____ by
substitution.
c) If ∠ P ≅ ∠ R and ∠ Q ≅ ∠ R , then ___ ≅ ∠ ___
1) Use a chain of reasoning to complete.
2)
P ( x+y+a)˚
R
( 2y +a)˚
Q
Express angle Q in terms of “x” and “a”
∠ P ≅ ∠ R
______________ = _______________
______________ = ______________
______ = y
Since m ∠ P = x + y + a we can substitute in the value equal to Y
m ∠ P = x + __ + a
m ∠ P = _______ therefore, since ∠ P ≅ ∠ R ,
∠ Q =________
K
Given: FG ≅ KJ
GH ≅ KJ
Prove: KG bisects FH
F
J
G
15
H
Statements
Reasons
Given : ∠ 1 + ∠ 2 = 90˚
∠ 1 ≅ ∠ 3
1
2
Prove : ∠ 3 + ∠ 2 = 90˚
Statements
3
Reasons
2.8 Vertical Angles
Definition # 19- 2 collinear rays that have a common endpoint and
extend in different directions are called opposite rays.
1) Draw: AB and AC are opposite ray 2)Then draw them on this segment
D
F
16
3) Now draw them as two rays that are not part of the same line.
4) PT and RS are not opposite because ________________________
__________________________________________________________
T
P
R
S
Vertical Angles:
Definition # 20: Two angles are vertical angles if the rays forming the
sides of one and the rays forming the sides of the other are opposite
rays.
2
1
3
4
Theorem # 18 - Vertical angles are congruent.
Which pairs of angles congruent.____________________________
If ∠ 1 = 2x + 5
and ∠ 3 = x + 30 , find the measure of each angle.
1
Given: ∠ 2 ≅ ∠ 3
2
Prove: ∠ 1 ≅ ∠ 3
3
Statements
Reasons
17
Given: ∠ O is comp to ∠ 2
∠ J is comp to ∠ 1
Conclusion: ∠ O ≅ ∠ J
O
H
1
J
Statements
2
Reasons
K
M