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Geometry
Topics Included:
1. Undefined Terms in Geometry
> have no exact definition, just a description
Point
* exact location/position on a plane surface
* represented by a dot & named with a capital letter A
Line
* straight, no thickness & extends in both directions w/o end
* set of infinite points

* named by any of its 2 points or italicized small letters
kA
B
line AB or line k
Plane
* flat surface that is infinitely large with zero thickness
* extends without end in all directions
* NO BOUNDARIES
* pictured by a 4 sided polygon
* contain infinitely many lines
* intersection of 2 distinct planes is always a line.
* intersection of more than 2 planes is SOMETIMES a line
* named by a italicized capital letter or 3 of its non-collinear points
D
Plane ABC or Plane D
Collinear points: set of points that lie in a straight line
Coplanar points: set of points that lie on the same plane
2. Lines & Line Segments
Segment: set of points on a line consisting of 2 end points and all the points between.
Ray: has one endpoint but the other side extends infinitely in one direction.
Opposite rays: collinear, have a common endpoint & extend in opposite directions
Betweenness (Segment Addition Postulate)
If BA + AC = BC, then A is between B and C.
Midpoint of a Segment
M is the midpoint of AB if:
* M lies between A and B
* AM = MB
Midpoint theorem: 2 AM = AB  AM = ½ AB
Every segment has exactly one midpoint
Segment Bisector
 Plane, line, segment or ray that intersects a segment only at its midpoint
DC is the bisector of segment AB
perpendicular bisector: bisector that is perpendicular (90 degrees) to the segment only at its
midpoint. (therefore DC is also a perpendicular bisector)
3. Angles
> union of 2 non-collinear rays which have a common endpoint
Right angle theorem: all right angles are congruent (equal to each other)
Angle bisector: a ray coplanar with the angle, which lies in the interior of the angle and divides
the angle into 2 congruent angles.
every angle has exactly one bisector
4. Angle Pairs
A. Adjacent Angles
> has a common vertex & common side, NO interior points in common.
Angle A and angle B are adjacent
B. Complementary Angles
> sum of measures is 90 degrees.
C. Supplementary Angles
> sum of measures is 180 degrees.
D. Linear Pair
> adjacent angles with a sum of 180 degrees
E. Vertical Angles
> non-adjacent angles formed by 2 intersecting lines
5. Relationships of Lines
Parallel lines: coplanar, non-intersecting. Even if you extend them, they will never meet.
segment A and B are parallel
Perpendicular lines: coplanar, intersecting & if extended, will form a right angle
If you extend line AB, it’ll intersect with line CD & form a right angle. Therefore they are
perpendicular.
CD
AB
Oblique lines: coplanar, intersecting & do NOT form a right angle when extended.
CD
EF
Skew lines: non-coplanar & non-intersecting. Usually one is horizontal & the other is vertical
PS and ZY are skew lines. Even though it LOOKS like they can intersect if extended, they don’t
meet because they aren’t coplanar.
6. Angles formed by a Transversal
Transversal: intersects two or more coplanar lines at two or more points
Exterior angles
Interior angles
Exterior angles
A. Alternate Interior Angles
* non-adjacent interior angles that on opposite sides of the transversal
* if lines are parallel, alternate interior angles are CONGRUENT according to PAIAT
(Parallel-Alternate Interior Angle Theorem)
Ex. <4 and <6
<5 and <3
B. Alternate Exterior Angle
* non-adjacent exterior angles that lie on opposite sides of the transversal
* if lines are parallel, alternate exterior angles are CONGRUENT according to PAEAT
(Parallel-Alternate Exterior Angle Theorem)
Ex. <1 and <7
<2 and <8
C. Corresponding Angles
* non-adjacent angles (one interior & one exterior) and are on the same side of the
transversal
* if lines are parallel, corresponding angles are CONGRUENT according to PCAP
(Parallel-Corresponding Angle Postulate)
Ex. <1 and <5
<2 and <6
D. Interior Angles on the Same Side of the Transversal (SST)
* adjacent interior angles on the same side of the transversal
* if lines are parallel, interior angles on the SST are SUPPLEMENTARY according to
PIASST (Parallel-Interior Angles on the Same Side of the Transversal)
Ex. <4 and <5
<6 and <3
E. Exterior Angles on the Same Side of the Transversal (SST)
* adjacent exterior angles on the same side of the transversal
* if lines are parallel, exterior angles on the SST are SUPPLEMENTARY according to
PEASST (Parallel-Exterior Angles on the Same Side of the Transversal)
Ex. <1 and <8
<2 and <7
7. Polygons
 a closed plane figure formed by three or more line segments called sides.
PARTS:
> vertex: corner of a polygon
A, B, C, D, E and F are the vertices
> sides: segments connecting 2 consecutive
vertices.
AB, BC, CD, DE, EF and FA.
> angles: interior angles formed by the sides
of the polygon.
> diagonals: segments connecting 2 nonconsecutive vertices.
> interior points: points inside the polygon
> exterior points: points outside the polygon
Naming a polygon?
Vertices (capital letters) written consecutively in a clockwise or counterclockwise manner starting
at any letter.
Kinds of Polygons
Convex: all diagonals of the polygons are in the interior. If the sides are extended, they will never
intersect with another side.
Concave: one or more diagonals are in the exterior. If sides are extended, they intersect another
side.
8. Diagonals
n = number of sides of a polygon
d = number of diagonals per vertex
d = n -3
D = total number of diagonals of a polygon
dn n(n  3)
D

2
2
T = number of triangles formed with respect to a vertex
T n 2
SI = sum of measures of the interior angles of a polygon

SI  180n  2
 SE = sum of the measures of the exterior angles of a polygon
SE = 360
IR = measure of each interior angle of a regular polygon

SI 180(n  2)
IR  
n
n
ER = measure of each exterior angle of a regular polygon
SE 360
ER 

n
n


Please review the powerpoints posted on the Notes and HW site for problems about polygons and
diagonals. 