Download Geometry: Section 1.2 Start Thinking: How would you describe a

Geometry: Section 1.2
_______________________________________________
Start Thinking: How would you describe a point, line and plane. Remember, with only your description I should be
able to draw the figure.
Undefined Terms:
Name
Description
Diagram and Notation
Point
Line
Plane
Collinear
Coplanar
PRACTICE:
1a.) Rename Line M in 2 ways.
2a.) Name Plane N in 2 ways.
3a.) Name a point that is coplanar with A B C.
4a.) Name a point that is collinear with E and A
1b.)What point is coplanarr with C, B and G?
2b.) How many points are coplanar with C?
3b.) What points are coplanar with B, D and G?
Defined Terms:
Name
Description
Diagram and Notation
Segment
Ray
Opposite rays
PRACTICE:
1.) Name the Largest Segment in two different ways.
2.) Name
in two different ways.
3.) What two segments create
4.)What is an opposite ray with
5.)What is the overlap of
and
?
?
6.) Do rays with a common end point always create a line? Draw a diagram.
Definition:
Postulate/ Axiom: _______________________________________________________________________________
______________________________________________________________________________________________.
Postulate 1.1: Though any two points there exists exactly _______________ line(s).
Postulate 1.2: If two lines intersect, they intersect at _______________________________.
Postulate 1.3: If two planes intersect, then they intersect at __________________________.
Postulate 1.4: Through three noncollinear points there is exactly one ________________________.
PRACTICE:
1.)
and
intersect at ____________.
2.) Plane HEF and Plane DCE intersect at ______________.
3.) Plane HGB and
intersect at _________________.
4.) Plane FEC intersects Plane ADB at ____________________.
Sometimes, Always, Never
1.) Three points are ________________________ coplanar.
2.) Three points are _____________________ collinear.
3.) Two intersecting lines are ________________________ coplanar.
Determine the location where the following two lines intersect.
1.) Y = -3x + 8 and -2x + y = -2
2.)Y = 3 and y = -2/3x + 7
Reasoning:
Though there are not actually any truly real world examples of the geometric figures we have discussed, give examples
of figures that represent the following. Explain why they can’t actually be classified as the figure.
1.)Point:
2.) Line
3.)Plane
3.) Ray
Geometry: Section 1.3
***_______________________________________________***
Vocabulary:
Definition
Example and Notation
Line Segment:
Length of a Line Segment:
Postulate 1.6: If three points A, B and C are collinear and C is between A and B then __________________________.
Practice:
1.)
contains point J. If MN= 59 and MJ=8x-14 and JN= 4x +1 then what does x equal?
2.)
contains point G. If KG= 3x+2 and GM=2x-3 and KM= 4x +10 then what is KM?
Definition
Examples and Notation
Congruent:
Equal:
Midpoint:
Segment Bisector:
3.)
contains point J such that
4.) P is the midpoint of
. If MJ= 6x-7 and JN=5x+1 then what does x equal?
. If M is at -12 and N is at 2 where is P?
4.)
has M as its midpoint. If T is located at -4 on the number line and U is to the right of T, and TM = 8x+11
and MU=12x – 1, then where is U located?
6.)
intersects
at its midpoint K. If AK= 3x+10 and KB=46 then what does x equal?
intersects
. They both bisect on another and intersect at a 90 degree angle at P. If KP=3 and
MP = 3x+1 and PN=4x then what is the distance between M and K?
7.) Relate two objects to one another using the word “congruent” and then change the sentence to describe an aspect
of them using the word equal.
Geometry: Section 1.4
_______________________________________________
Vocabulary:
Definition
Example and Notation
Angle:
Measure of an Angle:
Ex1:
Name ALL the angles in the diagrams.
Term
Example
Acute angle: An angle that measure between _________________________
Right angle: An angle that measures ________________________
Obtuse angle: An angle that measures _____________________________
Straight angle: An angle that measures ____________________
Congruent angles:
Ex2: Identify congruent angles
Ex3: In the diagram below,
bisects LKN
and mLKM = 78°. Find mLKN.
If mP = 120°, what is mN? ______
If mL = 20°, what is mM?____
Ex3: Identify all pairs of congruent angles.
FINDING ANGLE MEASURES
Ex5: Use the diagram to find the measure of the indicated angle. Then classify the angle.
ANGLE ADDITION POSTULATE:
If P is in the interior of RST,
Ex6: Given that mGFJ = 155°, find mGFH
a right angle,
and mHFJ.
mRST = ________ + ________.
EX 7: Practice B:
Given that VRS is
find mVRT .
More examples
EX: <ABC is a straight angle. Point D is in the interior of the angle. If m<ABD is 3x -7 and m<DBC = 2x+2
then what is
m<ABD in degrees?
EX:
is bisected by
. If
= 8x -2 and
= 6x+22 then what is
EX:
Easier: John has a weird clock where the hour hand only moves at the top of the hour and does not
move as the hour is passing. What is the measure of the angle between the hour hand and the minute
hand at 12:20?
Harder: The hour hand of Danny’s clock does move as the minute hand moves. What would be the angle
between the hour hand and the minute hand at 12:30?
Geometry Lecture 1.5
______________________________
Word Bank:
Segment
Point
Congruent
Ray
Vertex
Equal
Line
Coplanar
Opposite Rays
Plane
Collinear
Right Angle
DIAGRAM
Adjacent angles ________________________________________________________________
_____________________________________________________________________________
Vertical angles _________________________________________________________________
_____________________________________________________________________________
Complementary angles __________________________________________________________
______________________________________________________________________________
Supplementary angles___________________________________________________________
______________________________________________________________________________
Linear pair_____________________________________________________________________
______________________________________________________________________________
Angle Bisector __________________________________________________________________
_______________________________________________________________________________
Ex 1: Identify an example of the following types of angles/ objects.
a.) Adjacent Angles
b.)Vertical Angles
c.)Supplementary Angles
d.)Complementary Angles
e.)Angle Bisector
f.) Linear Pair
Ex 2: Draw a figure with the following characteristics.
1.)∠ADB and ∠BDC are adjacent
bisects ∠ADC
∠EDF and ∠BDC are vertical angles.
EX 3: ∠BMC and ∠CMD are complementary. If m∠BMC = 6x +1 and m∠CMD = 4x+11. Solve for x.
EX 4: ∠ABC and ∠CBD are Supplementary. If m∠ABC = 20 and m∠CBD = 15x+10 . Solve for x.
EX 5:
bisects ∠ADC. If m∠ADB = 8x – 2 and ∠BDC = 6x + 38 then what is m∠ADC
Vertical Angles
Vertical Angles are ______________________________.
Example:
1.)
2.)
Challenge Practice:
1.) :
bisects ∠AFC . Solve for x.
2.) Find the values of x and y shown in the diagram.
Geometry: Section 1.7
_____________________________________________________
Vocabulary:
Length of a Line Segment: _________________________________________________________________________
THE DISTANCE FORMULA. The distance d between two points A(x1, y1)and B(x2, y2) is: ______________________________
***Developing the Distance formula***
Example: A(2, 9) B(-3, 10)
AB= ?
Ex 2: Given points J(6,6) and K(2,-4), find JK.
Ex 1: Luisa takes the subway from Oak Station to Jackson Station each morning. Oak Station is 1mi west and
2mi south of City Plaza. Jackson Station is 2mi east and 4mi north of City Plaza. City Plaza is at (0,0). Using the
distance formula, determine how far Luisa travels by subway.
Vocabulary:
Midpoint of a segment: ____________________________________________________________________________
THE MIDPOINT FORMULA: the formula for the location of the midpoint is
Ex 2: Given points J(6,6) and K(2,-4), find midpoint of
Ex 3: Use the midpoint formula to find the endpoint C.
If midpoint of
is M(3,4) and endpoint is A(1,6). Find the coordinate of endpoint C.
:
Ex 4: The Line Segment AB has J as its midpoint. If A (1, 5) and J (7, -7) what is the location of B?
Ex5:
has its midpoint at R. If we have the following locations, G(1,3) and R (4, 7) and RH = 2x+1 then what does x
equal?
EX 6:
Ex 7:
has a midpoint M.
has its midpoint at A. Determine the location of A if J (2,1) and K(12,7).
has a midpoint at L. If M is located at (0,0) and N is at (a,b) then determine the expression that describes the
distance between MN as well as the location of L.
EX 8: Flagstaff has decided to put a tightrope over the buildings in downtown. If they plan to have one
end at the corner of RT 66 and Humphrys and the other end is at san Francisco and Dale and a city block
is 1000ft, then how long will the tight rope be? If the tightrope walker walks at a rate of one mile per
hour, then how long will it take to walk the whole length to the nearest second?
Geometry Section 1.8
_________________________________________________
START THINKING: How would you describe area and perimeter? When might you need each?
Formulas:
Figure
Perimeter
Square
Rectangle
Triangle
Circle
Parallelogram
Let’s think:
Do you need to know area or perimeter to know how much to buy of:
a.
b.
c.
d.
Wallpaper for a bedroom
Fencing for a backyard
Crown molding for a ceiling
Paint for a basement floor
Area
.
Practice: Calculate the area and perimeter of the following.
1.)
3.)
2.)
4.)
Working Backwards
1.) What is the length of x given the figure below and that the area is 528 cm.
2.) Given an area, find the value of x.
A = 399 in.2
5.) Determine the area of the following composite figures.
a.)
b.) Not to Scale
KM
More Challenging Problems:
1.) The base of a triangle is four times its height. The area of the triangle is 50 square inches. Find the base and height.
2.) The length of a rectangle is 4 less than twice the width. If the perimeter is 48 in then what are
dimensions of the rectangle?