Transcript 1.2 Points, Lines, & Planes

1.1 Building Blocks of
Geometry
Geometry Terms
Definition: Known words used to describe a
new word.
Postulate: A statement that is accepted as
true without proof sometimes they are
called axioms.
Theorem: Important statements that are
proven.
Segment
1.
2.
3.
4.
5.
A
Begins at one point and ends at another
Has points on each end called endpoints
Consists of an infinite amount of points
Always straight
Named by its endpoints, in either order
B
Can be called:
Segment AB
Segment BA
AB or BA
Point
1. Has no dimension
(no length, width, thickness)
2. Usually represented by a dot
3. Named using one capital letter
• B
Line
1. Extends forever in one dimension (length)
2. Has an arrowhead on each end representing the fact
that it goes on forever
3. Consists of an infinite amount of points
4. Always straight
5. Named with a lowercase script letter or by two points
on the line
Can be called:
A
B
l
Line l
Line AB or AB
Line BA or BA
Plane
1.
2.
3.
4.
Extends forever in 2 dimensions (length & width)
A flat surface consisting of infinitely many points
Usually represented by a 4-sided figure
Named with a capital script letter or 3 noncollinear
points on the surface of the plane
W
A
B
C
Can be named:
Plane W
Plane ABC, plane BCA, plane CBA,
(any three noncollinear points)
Different planes in a figure:
A
D
B
C
E
H
Plane EFGH
F
G
Plane ABCD
Plane BCGF
Plane ADHE
Plane ABFE
Plane CDHG
Other planes in the same figure:
Any three non collinear points determine
a plane!
A
D
B
Plane AFGD
Plane ACGE
C
Plane ACH
E
H
F
G
Plane AGF
Plane BDG
More Definitions
Collinear points – points that lie on the same line
More Definitions
Coplanar points: points that lie on the same plane
Coplanar lines: lines that lie on the same plane
A, B, and C are coplanar points
Lines l and n are coplanar lines
Is Alex between Ty and Josh?
Yes!
Ty
Alex
Josh
How about now?
No, but
why not?
In order for a point to be between two
others, all 3 points MUST BE collinear!!
Ray
• Piece of a line with only one endpoint (initial point)
and continues forever in the other direction
A
B
• Named by the endpoint and a second point named
on the ray. (name MUST begin with the endpoint!)
AB
Opposite Rays
• Two rays that share a common initial point
and face opposite directions.
P
Q
S
• QP and QS are opposite rays.
More Definitions
• Intersect – two or more figures intersect if they
have one or more points in common.
• Intersection – all points or sets of points the figures
have in common
• What is the intersection of:
AB & DA
BC & AC
BC & BC
When two lines intersect, their
intersection is a point.
When two planes intersect,
their intersection is a line.
B
P
A
R
Plane P and Plane R intersect at the line AB
Angle symbol: 
• Two rays that share the same endpoint (or initial
point)
Sides – the rays XY & XZ
Y
5
X
Z
Named YXZ, ZXY (vertex is
always in the middle), or X (if it’s
the only X in the diagram).
Vertex – the common endpoint;
X
Angles can also be
named by a #. (5)
There are 3 different B’s in this diagram;
therefore, none of them should be called B.
A
B?
D
B
C
Interior or Exterior?
• B is ___________
in the interior
• C is ___________
in the exterior
on the A
• D is ___________
B
C
D
A
Assignment
Section 9 - 47
1.2 Measuring Length
Ruler postulate
• The points on a line can be matched with
those on the real number line.
• The real number that corresponds to a
point is the coordinate of the point.
• If you find the difference between the
coordinates of two points, then take the
absolute value, you will have the
distance or length between the points.
Ruler postulate (continued)
A
B
mAB = AB =a – b or b – a
The symbol for the length of AB is AB.
Example: Find AB.
A
Point A is at 1.5 and B is at 5.
So, AB = 5 - 1.5 = 3.5
B
Determine the length of a given segment.
Find the lengths of AB, BC, and CD.
AB = – 4 – (–1)= 3
BC = –1 – 4= 5
CD = 4 – 9= 5
Determine whether segments are congruent.
In the figure, BC  CD, but AB is not congruent
to the other segments.
Segment Congruence Postulate
Segment Congruence Postulate: If two segments have
the same length then the segments are congruent.
Also if two segments are congruent then they have the
same length if measured by a fair ruler.
If AB & XY have the same length,
Then AB = XY,
and
AB  XY
Symbol for congruent
Segment Addition Postulate
If B is between A & C, then AB + BC = AC.
If AB + BC = AC, then B is between A & C.
A
B
C
Example: If DE = 2, EF = 5, and DE = FG,
find FG, DF, DG, & EG.
D
E
F
FG = 2
DF = 7
DG = 9
EG = 7
G
Questions
Assignment
Practice A, B and
Section 11 - 27
1.3 Measuring Angles
A protractor is a device used for
measuring angles. As on a ruler the
intervals on a protractor are equal.
Angle Measurement
• mA means the
“measure of A”
• Measure angles with
a protractor.
• Units of angle
measurement are
degrees (o).
• Angles with the
same measure are
congruent angles.
• If mA = mB,
then A  B.
Measure of an Angle
• The rays of an angle
can be matched up
with real numbers
• (from 0 to 180) on a
protractor so that the
measure of the 
equals the absolute
value of the
difference of the
numbers.
55o
20o
mA = 55 - 20
= 35o
Determine the measure of a given angle.
Find the measures of angle BVC.
m BVC =125 – 50= 75
Add measures of angles.
m AVC = m AVB + m BVC
= 25 + 75 = 100
Angle Addition Postulate
If P is in the interior of RST,
then mQRP + mPRS = mQRS.
If mQRP = 5xo,
S
P
mPRS = 2xo, &
mQRS = 84o, find x.
Q
5x + 2x = 84
7x = 84
x = 12
R
mQRP = 60o
mPRS=24o
Angle Congruence Postulate
If two angles have the same measure,
then they are congruent. If two angles are
congruent, then they have the same
measure.
Types of Angles
• Acute angle –
Measures between 0o & 90o
• Right angle –
Measures exactly 90o
• Obtuse angle – Measures between 90o & 180o
• Straight angle – Measures exactly 180o
Adjacent Angles
Two angles that share a common vertex & side,
but have no common interior parts.
(they have the same vertex, but don’t overlap)
such as 1 & 2
2
1
Example:
• Name an acute angle
3, 2, SBT, or TBC
• Name an obtuse angle
ABT
• Name a right angle
1, ABS, or SBC
• Name a straight angle
ABC
S
T
3
1
2
A
B
C
Midpoint
• The point that bisects a segment.
• Bisects?
splits into 2 equal pieces
12x + 3
A
10x + 5
M
12x + 3 = 10x + 5
2x = 2
x=1
B
Segment Bisector
A segment, ray, line, or plane that
intersects a segment at its midpoint.
k
A
M
B
Angle Bisector
A ray that divides an angle into two congruent
adjacent angles.
A
D
B
C
BD is an angle
bisector of ABC.
Example: If FH bisects EFG and
mEFG = 120o, then what is mEFH?
E
120
o
 60
2
H
F
G
mEFH  60
o
Example: Solve for x.
* If they are
congruent, set them
equal to each other,
then solve!
x + 40 = 3x - 20
40 = 2x - 20
60 = 2x
30 = x
Which angles are adjacent?
1 & 2, 2 & 3, 3 & 4, 4 & 1
Then what do we call 1 & 3?
2
1
3
4
Vertical Angles – two angles
that share a common vertex &
whose sides form 2 pairs of
opposite rays.
1 & 3, 2 & 4
Linear Pair
A linear pair is two adjacent angles whose
non-common sides are opposite rays.
These angles form a straight line and their
sum is 180°.
Example
• Vertical angles?
1 & 4
• Adjacent angles?
1 & 2, 2 & 3,
1
5
3 & 4, 4 & 5, 5 & 1
• Linear pair?
5 & 4, 1 & 5
• Adjacent angles not a linear pair?
1 & 2, 2 & 3, 3 & 4
2
3
4
Important Facts
• Vertical Angles are congruent.
• The sum of the measures of the angles in
a linear pair is 180o.
Example:
If m5 = 130o, find
o
=130
m3
m6 =50o
m4 =50o
4
5
3
6
A
Example:
Find x and y
mABE
mABD
mDBC
mEBC
E
B
D
C
x = 40
y = 35
mABE = 125o
mABD = 55o
mDBC = 125o
mEBC = 55o
Complementary Angles
• Two angles whose sum is 90o
35o
1
2
1 & 2 are complementary
A & B are complementary
55o
A
B
Supplementary Angles
Two angles whose sum is 180o
1 & 2 are
supplementary.
X & Y are
supplementary.
130o
X
50o
Y
Example: A & B are supplementary.
mA is 5 times mB.
Find mA & mB.
mA + mB = 180o
mA = 5(mB)
Now substitute!
5(mB) + mB = 180o
6(mB)=180o
mB=30o
mA=150o
Perpendicular Bisector
A perpendicular bisector intersects at the
midpoint AND is perpendicular to the segment.
Parallel Lines
Parallel Lines
Two lines are parallel lines if they lie in the same
plane and do not intersect.
Perpendicular Lines
Two lines are perpendicular lines if
they intersect to form right angles.
Skew Lines
Skew are lines that do NOT lie in the same
plane and do NOT intersect.
Questions
Find the measure of each of the angles.
Questions
Questions
Questions
Assignment
Practice B and
Section 14 - 44