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Geometry Definitions, Postulates, and Theorems
Chapter 1: Essentials of Geometry
Section 1.1: Identify Points, Lines, and Planes
Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms,
theorems, and inductive and deductive reasoning.
Three Undefined Terms in Geometry: *Point, Line, & Plane. Descriptions are below.
Point –
Line –
Plane –
Collinear –
Ex. Points Q, R, and T are collinear.
Points Q, R, and S are noncollinear.
Coplanar – Points that lie on the same plane.
Ex. Name 3 points that are coplanar 
Name 4 points that are noncoplanar 
Name 3 collinear points 
Segment – Symbolized AB .
(over)
Ray – Symbolized AB.
Opposite Rays –
Intersect –
Ex. Draw a line and a plane that do not intersect.
Ex. Draw 2 lines that intersect and one that does not.
Intersection –
Ex. Draw a line and a plane that intersect. Their intersection forms a
Ex. Draw 2 planes that intersect. Their intersection forms a
Ex. Draw 3 collinear points A, B, C.
Draw D which is noncollinear with A, B and C.
Draw AB and BD .
.
.
Section 1.2: Use Segments and Congruence
Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms,
theorems, and inductive and deductive reasoning.
Postulates or Axioms –
Coordinate –
Distance of a segment –
Ruler Postulate –
Between –
***Segment Addition Postulate – IF B is between A and C, THEN AB + BC = AC.
IF AB + BC = AC, THEN B is between A and C.
Ex. Given GK = 24, HJ = 10, GH=HI=IJ,
find HI and IK.
G
H
I
J
K
***Congruent Segments –
Ex. J is between H and K. Solve for x.
HJ = 2x + 5, JK = 3x – 7, KH = 18
Section 1.3: Use Midpoint and Distance Formulas
Standards: Prepare for 7.0 Students prove theorems by using coordinate geometry, including the midpoint of a line
segment, the distance formula, and various forms of equations of lines and circles.
***Midpoint –
Bisects –
***Segment Bisector –
Ex. M is the midpoint of segment GH . Find MH.
G
M
5x – 8
H
3x + 14
Midpoint Formula –
Ex. Find the coordinates of the midpoint of
AB if A(-2,-3) and B(5,-2).
Ex. The midpoint of JK is M(0, ½).
One endpoint is J(2,-2), find K(x,y).
(over)
Distance Formula –
Ex. Given A(3,2) and B(-1,1), find AB.
Ex. Find the distance between B (-1, 1) and C(-2, -1).
Pythagorean Theorem –
c
a
a 2 + b 2 = c2
b
Review: Simplifying Radicals. (Doubles get you out of jail)
Ex.
a)
20
d)
80
b)
c)
60
e)
1050
72
Section 1.4: Measure and Classify Angles
Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms,
theorems, and inductive and deductive reasoning.
Angle –
Sides –
Vertex –
An angle is symbolized by three letters, where the center letter is the vertex.
Measure of an angle –
Protractor Postulate –
Acute Angle –
***Right Angle –
Obtuse Angle –
(over)
***Straight Angle –
***Angle Addition Postulate –
***Congruent Angles –
Ex. Find mSTV .
S
Ex. Given mMNP  70 ,
find mMNO .
P
Ex. Find the missing angle.
U
O
30°
20°
80°
120°
N
?
T °
V
M
***Angle Bisector –
Ex. CD bisects ACB . Given mACD  ( x  10) and mBCD  (3 x  20) , solve for x and find
mACB .
Section 1.5: Describe Angle Pair Relationships
Standards: 13.0 Students prove relationships between angles in polygons by using properties of complementary,
supplementary, vertical, and exterior angles.
***Complementary Angles –
Complement –
Ex. A and B are complementary. Given mA  (5 x  8) and mB  ( x  4) , find mAand mB .
***Supplementary Angles –
Supplement –
Ex. A and B are supplementary. Given mA  (3x) and mB  ( x  8) , find mAand mB .
Adjacent Angles –
***Linear Pair –
1
4
3
2
(over)
***Vertical Angles –
1
4
3
2
Ex. True or False.
a) 1 and 2 are a linear pair.
2
b) 4 and 5 are a linear pair.
1
c) 5 and 3 are vertical angles.
5
3
4
d) 1 and 3 are vertical angles.
e) If m3  40 , find: m1 
m2 
m4 
m5 
Ex. Given m1  (4x + 15)°, m2  (5x + 30)°, m3  (3y – 15)°, and m4  (3y+15)°, find x and y
and the measure of each angle.
1
4
2
3
Section 1.6: Classify Polygons
Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to
classify figures and solve problems.
Polygon – A plane figure that:
1.
2.
Sides –
Vertex –
Ex. Is it a polygon? (write yes or no)
a)
b)
c)
d)
e)
Convex –
ENGLISH TRANSLATION – A polygon without any dents!
Nonconvex (or concave) –
ENGLISH TRANSLATION – A polygon with a dent, or two, or three, etc.!
Polygons are named by the number of sides they have.
Number of sides
3
4
5
6
7
8
9
10
11
12
n
Type of polygons
triangle
quadrilateral
pentagon
hexagon
heptagon or septagon
octagon
nonagon
decagon
hendecagon
dodecagon
n-gon ("n" sides, where n is a variable)
(over)
Regular –
1.
2.
Diagonal –
Ex. Draw a regular hexagon. Draw all the diagonals.
Ex. Find the value of x in the regular polygon.
x2 + 3x – 4
x2 – x
Section 1.7: Find Perimeter, Circumference, and Area
Standards: 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral
area, and surface area of common geometric figures. 10.0 Students compute areas of polygons, including rectangles,
scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
Perimeter –
Area –
Square –
Ex.
3 cm
Perimeter =
Area =
Perimeter =
Area =
Perimeter =
Area =
3 cm
Rectangle –
8”
Ex.
5”
Triangle –
Ex.
8mm
9mm
6mm
11mm
Circle –
Ex.
4cm
Radius –
A=
Diameter –
Circumference –
C=
Pi – Symbolized  .
(over)
Ex. Find the area and perimeter of the triangle defined by H(-2,2), J(3,-1) and K(-2,-4).
y
x
Ex. A rectangle has an area of 15.4 ft2. It’s length is 7 ft. What is the width?