Download Chapter 1 Notes 2013

Name _____________________________
Geometry
Date _________
Period _________
1) Get rid of parentheses by distributing
2) Combine like terms that are on the same side of the equal sign
3) Move all the variables to one side of the equation.
4) Move all the constants to the other side of the equation.
5) Solve like normal.
Examples:
1) n+12 = 25
2) r- 14 = 13
5) 2y+1=15
6)
9) 3y-7y=28
h
 3
15
3) 9x=72
4)
1
x3  5
2
7) -3(y-5)=12
8) 2(5x+4)=38
10) 4c+3(c-2)= -34
11) 6x=4x+18
12) 4n+5=6n+7
13) The length of a rectangle is 25 ft more than the width. The perimeter is 325 ft.
14) The width of a rectangle is 15 cm less than the length. The perimeter is 98 cm. Find the rectangle’s
dimensions.
1
Name _____________________________
Geometry
Date _________
Period _________
Coordinates
– ordered pair ___________
– _______________________________
 x – 1st coordinate
(____________________________)
 y – 2nd coordinate
(____________________________)
Space – set of all points
- contains all _______, _______, and _________
2
Name _____________________________
Geometry
Defining Geometric Vocabulary:
Date _________
Period _________
1) State the term
2) State the nearest classification
3) Describe what makes it unique.
Examples:
Collinear –
Ex:
C
B
Noncollinear –
D
A
Ex:
Coplanar –
Ex:
Noncoplanar –
Ex:
Points, Lines, & Planes: CLASS WORK
Use this diagram
for #1-5
1. Name three collinear points on line q and on line s
H
q
I
2. Name 1 set of non-collinear points
L
3. Name the opposite rays on line p and on line s
4. How many points are marked on line q?
K
J
s
5. How many points are there on line q?
21. Name a point that is coplanar with A, E, and J
A
22. Name a point that is coplanar with A, C, and I
C
23. Name all the points that are noncoplanar with A, C, and D
24. Name all the points that are noncoplanar with F, H, and E
F
H
25. Where do plane ACH and plane IDC intersect?
26. Where do planes ACH, AFJ, and ACD intersect?
E
D
3
J
I
#21-32
Name _____________________________
Geometry
Activity: Developing Definitions for One-Dimensional Figures
Date _________
Period _________
Midpoint:
Example:
Counterexample:
Parallel Lines:
Example:
Counterexample:
Congruent Segments:
Example:
Counterexample:
Bisector of a Segment
Example:
Counterexample:
C
Segment – Labeled as _________________________
- contains all points between the two end points.
B
A
Q
Measurement of a segment – distance between ________________
- measured in units of inches, feet, yards, meters, cm… etc.)
#7) BD=
#8) DA=
#9) AC=
#10) CD=
4
Name _____________________________
Geometry
Parts of a Right Triangle:
Date _________
Period _________
 Hypotenuse
 Leg a and leg b
 Angle opposite hypotenuse
 Side opposite right angle
 Longest side
______________________________:
In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the
hypotenuse.
Formula:
c
a
b
Note: _______ is always the ______________________
Example #1)
Example #2)
x
13
7
x
24
5
Using the Pythagorean Theorem to find the Distance in the Coordinate Grid:
#1)
#2)
5
Name _____________________________
Geometry
#3)
Date _________
Period _________
Distance Formula
 What is the formula?
 When do you use it?
Examples:
#1) Find the distance between W and Z, when W(1, 2) and Z(-4, -2)
#2) Find WY.
#3) Find XZ.
A
6
Name _____________________________
Geometry
Date _________
Period _________
How do you find the midpoint of a line?
A
B
C
D
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Example #3) Find the midpoint.
a) BD
b) AB
c) DA
d) CD
Example #4)
Midpoint Formula:
 What do you use it for? To find the midpoint of a line segment on the
______________________________________.
Examples:
#1) Find the midpoint of (-1, 7) and (2, 5)
#2) Find the midpoint of (3, -5) and (7, 1)
#3) Find the midpoint of (-1, 4) and (-1, 6)
7
Name _____________________________
Date _________
Geometry
Period _________
#4) M is the midpoint of CD, where C(3,-2) and D(x, y) and M(-3, 1). Find the coordinates of D.
#5) If N(4, -1) is the midpoint of ST and the coordinates of S are (3, -2), what are the coordinates of T?
____________________- How rapidly the line of an equation rises (positive slope) or declines (negative slope)
Formula: m=
Directions: Determine the slope of each line.
a)
b)
c)
d)
8
Name _____________________________
Geometry
Date _________
Period _________
A ____________________ is a comparison of two quantities
The ratio of a to b can be expressed as:
Connor has a wallet with:1-$20 bill 2- $10 bills
1- $5 bill
8-$1 bills
1) What is the ratio of $1 bills to $10 bills?
2) What is the ratio of $10 bills to the total number of bills in the wallet?
Point P divides AB in the ratio 3 to 1.
1. What does this mean?
Example: A 32 foot long piece of rope has a knot tied to divide the rope into a ratio of 3:5.
Where should the knot be tied?
9
Name _____________________________
Date _________
Geometry
Period _________
Directed Line Segment: Tells the direction in which from which point to start and end.
In this case, from Point A to Point B
What does that tell you about the distance AP and PB in relation to AB?
Example 1: Find the coordinate of point P that lies along the directed line segment from A(3, 4) to B(6, 10) and
partitions the segment in the ratio of 3 to 2.
1. Find the rise and run for AB
2. Multiply the rise by the ratio from A to P, and
the run by the ratio of A to P
3.
Add/subtract these values to your starting point A
4. How can you use the distance formula to check that P partitions AB in the ratio of 3 to 2?
Example 2: Find the point Q along the directed line segment from point R(–3, 3) to point S(6, –3) that divides
the segment into the ratio 2 to 1
10
Name _____________________________
Date _________
Geometry
Period _________
Example 3: Find the point Q along the directed line segment from point R(–2, 4) to point S(18, –6) that divides
the segment in the ratio 3 to 7.
Example 4: Find the coordinates of the point P that lies along the directed segment from A(1, 1) to B(7, 3) and
partitions the segment in the ratio of 1 to 4
Example 5: Find the coordinates of point P that lies along the directed line segment from M to N and partitions
the segment in the ratio of 3 to 2
11
Name _____________________________
Geometry
Date _________
Period _________
Segment Postulate: If Q is between P and R, then ________________.
- If PQ + QR = PR, then _____ is between P and R.
Ex 1) Find x
Given:
A
AB = 20
Z
B
AZ = 8 + x
ZB = 5x
2) K, M, and P are collinear with P between K and M, PM = 2x + 4, MK = 14x - 56, and PK = x + 17
Solve for x and then find MK.
3) P, B, L, and M are collinear and are in the following order: P is between B and M, L is between M and P
Draw a diagram and solve for x and then find PL, given:
ML = 3x +16, PL = 2x +11, BM = 3x +140, and PB = 3x + 13
12
Name _____________________________
Geometry
What notation do I use for angles?
Date _________
Period _________
S
vertex
Labeled as:
1)
1
P
2)
T
vertex: _______________________
3)
Why can’t I use one letter?
R
To call it < B is too confusing!!
S
___________________ – they’re 2 different <’s!
Sides of an angle – rays of angle
B
T
Ex. 1)
a.) What number name is <ABD?
D
A
1
b.) What is the vertex of < 3?
2
3
C
B
c.) What are the sides of < 2?
3 parts of an angle:
L
1)
2)
W
3)
M
X
N
13
Name _____________________________
Geometry
Date _________
Period _________
Measure of an angle – ____________
Use a protractor to measure an angle
Ex 1: A is in the interior of <BNJ. If m<ANJ=7x+11, m<ANB=15x+24, and <BNJ=9x+204. Solve for x and
find <ANJ.
14
Name _____________________________
Geometry
Example #3) Use a protractor to find the measure of each numbered angle.
3
2
Date _________
Period _________
1
4
______________ – form straight line
*****Opposite rays form an angle that is ______°
Example #2) In the figure, KZ and KX are opposite rays. Find the value of x and the measure of the indicated
angle.
m< XKP = 5x + 2,
m<PKQ = 3x + 4,
m< XKQ = 150; m< XKP
Q
P
X
K
Z
Examples:
#4) Find <RXT if <PXR=3x and <RXT=5x+20
S
Q
R
P
X
T
15
Name _____________________________
Geometry
#5) Find <QXT if QXT  2 x  75 and QXP  9 x  6
Date _________
Period _________
S
Q
R
P
X
T
What is a midpoint?
Midpoint – _____________
def: the midpoint, M, of PQ is the point between P and Q such that
___________.
P
M
Q
Congruent– exact same _________________ and ___________________
H
_______ is the midpoint of GH
G
J
GJ  JH
B
Example #1)
4
ft
A
X
4
ft
Y
Ex #2)
16
Name _____________________________
Geometry
Ex #3)
Date _________
Period _________
What happens if you cut an angle in ½?
E
 Get ________________________
 This is called the _______________________
D
F
___________________ – 2 congruent segments (midpoint)
___________________ – 2 congruent angles (bisecting ray must be
in the interior of the angle)
G
DF bisect <EDG
<EDF = <FDG
Example #1) Point D is in the interior of <ABC. Find <ABC if <ABD  <DBC, <ABD=11x-13,
and <DBC=5x+23.
Example #2) In the figure AM bisects <LAR. Find LAR if <MAR=2x+13 and <MAL=4x-3
R
S
A
M
L
17
Name _____________________________
Date _________
Geometry
Period _________
Example #4) BS bisects <ABT and AB bisects <RBT. Find the value of x and the measure of the indicated
angle.
a.) <RBA = 2x + 13, <TBA = 4x – 3; <RBT
A
R
S
B
T
b.) <TBS = 25 - 2x, <SBA = 3x + 5; <TBS
A
R
S
B
T
Adjacent angles – common __________________, common _______________, no same interior points (next
to)
Vertical angles – 2 nonadjacent angles formed by intersecting lines (opposite)
Vertical Angles:
Adjacent Angles:
R
1
4
Y
T
2
3
P
S
Vertical angles are ___________________!!!!
18
Name _____________________________
Date _________
Geometry
Period _________
Linear pair – two adjacent angles that form a _________________________________.
R
___________________________________ are
Y
a linear pair.
T
Sum of the measures of the angles in a linear
P
S
pair is ____________
Supplementary angles – sum of measures of 2 angles is _________
C
180 - x
x
A
B
D
Ex: Two angles are supplementary. One angle is twice as big as the smaller angle. What is the measure of both
angles?
Complementary angles – sum of measures of 2 angles is ____________
F
G
x
90 - x
E
H
Ex: An angle is 68˚ more than its complement. What is the angle’s measure?
More Examples:
G
#1) GH and LM intersect at Q. Find the value of x and the measure of <LQH.
L
Q
3x + 22
6x - 8
H
M
19
Name _____________________________
Geometry
#2) If <NOP and <POT are supplementary, find m<POS and m<SOT.
P
x + 28
90
N
Date _________
Period _________
S
6x - 15
O
T
#3) If <NPO and <NPM are complementary, find m<NPM and m<NPO.
M
N
26 + x
x
O
P
#4) In the figure, line MN and line OP intersect at D. Find the value of x and <ODN.
M
O
9x-4
P
D
4x-11
N
#5) If <BCE and <ECD are supplementary, find <ECF and <FCD.
F
E
63°
B
11x+2
5x-13
c
D
20