Download Geometry Non-Completed Notes

1-1 Understanding Points, Lines, and Planes
Point –
Line –
Plane –
Collinear – Points that lie on the same line.
Non-collinear – Points not on the same line.
Coplanar – Points or figures that lie on the same PLANE.
Non-coplanar – Points or figures not on the same PLANE.
K, L, M
KM
ABCD
line AE, line BE, line CE
Segment –
Endpoint –
Ray –
Opposite Rays –
Example C – Draw and label the following.
1) A segment with endpoints C and M.
C
M
2) Opposite rays with common endpoint S.
R
S
T
3) A line that intersects a plane but does not lie in the plane.
Postulate (axiom) – A statement that is accepted as true without proof.  1-1-1
Through any two points there is exactly one line.
 1-1-2 Through any three noncollinear points there is exactly one plane
containing them.
 1-1-3 If two points lie in a plane, then the line containing those points lies in
the plane.
 1-1-4 If two lines intersect, then they intersect in exactly one point.
1-1-5 If two planes intersect, then they intersect in exactly one line.
1-2 Measuring and Constructing Segments
The distance between any two points is the absolute value of the difference of the
coordinates. (Distance is ALWAYS positive!)
 Length of a Segment
 Example A – Find each length.
 1) AB = -2 – 1 = -3 = 3
 2) AC = -2 – 3 = -5 = 5
 3) BC = 1 – 3 = -2 = 2
Congruent Segments – segments that have the SAME length but different locations.
A
B
Segment AB is congruent to segment CD.
C
D
Segment Addition Postulate – If B is between A and C, then AB + BC = AC
3 + 2 = 5

 Example B
 1) G is between F and H, FG = 6, and FH = 11. Find GH.
FG + GH = 11
6 + X = 11
F
G
H
X = 5
GH = 5
 2) M is between N and O. Find x.
NM + MO = NO
17 + 3x -5 = 5x + 2
- (3x)
= - (3x)
12 = 2x + 2
10 = 2x
5= x
Midpoint – The point that divides (or bisects) the segment into congruent segments.
Segment Bisector – A ray, line, or segment that divides another segment through its midpoint.
 3) D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF.
4x + 6
7x - 9
ED = DF =26
4x + 6 = 7x - 9
- 4x + 9 - 4x + 9
EF =52
AB =
15 = 3x
5=x
AB
We use = when dealing with
We use
when dealing with
______NUMBERS______ .
____FIGURES__________ .
Review Over Section 1-1
E
F
1) How many planes are shown in the figure? 6
D
C
2) Are points B, E, G, and H coplanar? Explain. yes
Y
H
G
A
3) Name a point copl anar with D, C,. and B A
B
X
4) Name a line that contains point A. LINE l
5) What is another name for line m?
BD
6) Name a point not on AC .
E, D
D
l
A
B m
E
C
P
7) Name the intersection of AC and DB .
B
8) Name a point not on line l or line m. Explain.
E
1-3 Measuring and Constructing Angles
Angle – A figure formed by two rays, or sides, with a common endpoint, a vertex.
Right – an angle with a measure that is equal to 90 degrees.
Acute – an angle with a measure that is less than 90 degrees.
Obtuse – an angle with a measure between 90 and 180 degrees.
Ways to Name the Angle:
<R, <1, <SRT, TRS
Ways to Name the Angles:
<BAC, <CAB, <CAD, DAC, <BAD, DAC
Example A
Classify each angle as right, acute, or
obtuse.
1) < MPN
2) <NPR
ACUTE
RIGHT
3) < SPR
ACUTE
Congruent Angles –
angles that have the same measure.
4) <RPM
OBTUSE
Angle Addition Postulate –
<ADB +<BDC = <ADC
”Little” + “Little” = “Big”
Example B: m<DEG = 115°, and
Example C: m<XWZ = 121°
M<DEF = 48°. Find m<FEG
and m<XWY = 59°. Find m<YWZ
.
59
115
48
<DEF + <FEG = <DEG
<XWY + <YWZ = <XWZ
48 + x = 115
59 + x = 121
X = 67
X = 62
121
48
Angle Bisector – A ray that splits, or divides an angle into two
congruent angles.
<LJK = <KJM
JL bisects <LJM
Example D: KM bisects < JKL, m< JKM = (4x + 6)°, and m <MKL = (7x – 12)°.
Find x.
4x + 6 = 7x – 12
-4x + 12 = -4x +12
18 = 3x
6=x
Example E: QS bisects <PQR,
Example F: JK bisects <LJM,
M<PQS = (5y – 1)°, and m<PQR = (8y +
12)°. Find m<PQS.
M<LJK = (-10x + 3)°, and
m<KJM = (–x + 21)°. Find m<LJM
S
P
5y-1 + 5y -1 = 8y +12
10y -2 = 8y + 12
-8y +2 -8y + 2
Q
R
2y = 14
Y=7
-10x + 3= -x + 21
10x -21 10x -21
L
P
J
Q
K
S
-8y
M
R
-18 = 9x
-2= x
1-4 Pairs of Angles
Adjacent Angles – two angles that share a side.
They are “
NEXT TO EACH OTHER”
Linear Pair – two angles that are adjacent to each
other that form a line. (they sum 180 degrees)
<5 + <6 are a linear pair and sum to 180 degrees.
Example A – Tell whether the angles are adjacent, a
linear pair, both, or neither.
 1) <AEB and < BED
 2) <AEB and <BEC
 3) <DEC and <AEB
ADJ. & L.P.
ADJ.
NEITHER
Vertical Angles – created by two intersecting lines
Opposite angles are congruent, Adjacent
angles form a Linear Pair
<1 & <3
and <2 & <4 are Vertical Angles
Example B:
Identify each pair of angles as adjacent, vertical, and/or as a linear pair.
1) <1 and <2
ADJACENT
2) <1 and <6
ADJACENT & LINEAR PAIR
3) <1 and <5
4) <2 and <3
VERTICAL
ADJACENT
Complementary Angles –
Supplementary Angles –
Two angles that sum 90 degrees.
Two angles that sum 180 degrees.
Example C – Find the complement.
2)
90 – 59= 31
90 – (7x -12)
102 – 7x
Example D – Find the supplement.
1)
2)
180 – (7x + 10)
180 – 116.5
63.5
170 – 7x
Example E - An angle is 10° more than 3 times the measure of its complement. Find
the measure of the complement.
Example F –
1) If m<4=3x+7 and m<2=6x-5, find x.
Vertical Angles
3x +7 = 6x – 5
12 = 3x
4=x
M<2 = M<4 = 19
If m<3=6x+67 and m<2=9x-17, find x.
6x + 67 + 9x -17 = 180
x = 130 = 26
15
3
1-6 Midpoint and Distance in the Coordinate Plane
Midpoint – Think Average
Number Line a + b
2
Order Pairs { x1 + x2 , y1 + y2}
2
2
Example A – Find the midpoint of each segment.
1) CE = -4 +0 = -2
2)
DG= -1 + 8 = 3.5
2
2
3)
AF = -9 + 3 = -3
4)
EG = 0- 8 = -4
2
5)
2
AB = -8
6)
BG = -.5
Example B – Find the midpoint of each pair of ordered pairs.
1) A(0,0), B(12,8)
2) R(-12, 8), S(6,12)
0 + 12 , 0 + 8
2
2
-12 + 6 , 8 + 12
2
2
( 6, 4)
(-3, 10)
3) M(11,-2), N(-9,13)
4) J(2,5), K(2, -4)
11 + -9 , -2 + 13
2
2
2 + 2 , 5 + -1
2
2
( 1, 2)
(2, .5)
Example C – Given the midpoint, M of AB and one endpoint, find the missing endpoint.
1) B(2,7) and M(6,1)
2) A(-11, 2) and M(-15,4)
+4
+4
(2, 7)
(6, 1)
-5
(10, -5)
-5
-4
-4
(-11, 2)
(-15, 4)
(-19, 6)
+2
+2
Distance Formula –
Example D – Find the distance between the given two points.
1) A(0,0), B(15,20)
2) O(-12,0), P(-8,3)
(15 – 0)2 + (20 – 0)2
(-8 – (-12))2 + (3 – 0)2
152 + 202
42 + 3 2
225 + 400
16 + 9
625
= 25
25
=5
Example E – Determine if EF =GH .
E (-2, 1) F(-5, 5) d=5
G (-1, -2) H(3, 1) d=5
Pythagorean Theorem and the Distance Formula –
a2 + b2 = c2
32 + 42= c2 9 + 16 = c2
25 = c2 5 = c
Example F – Find the distance between the two points on the line.
1)
2) 8.06
a2 + b2 = c2
62 + 82= c2 10 = c
36 + 64 = c2
100= c2