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Geometry Notes
Chapter 1:
1.2 – Points, Lines, and Planes
Undefined Terms:
Point:



Line:

No size .
Diagram: A
Name: point A



Plane:

Extends in both
directions without
ending
At least 2 points
Name: EF, FE
Diagram:



Extends in all
directions without
ending
No thickness
Name: plane M
Diagram:
F
E
M
M
Space: set of all points
Collinear points:
Points all in ONE line
Note:
Lines intersect in a point
Planes intersect in a line
Non-collinear points:
Points that NOT all in ONE line
Coplanar Points:
Points all in ONE plane
Non-Coplanar Points:
Points NOT all in ONE plane
1.3 – Segments, Rays and Distance
Between: B is between A and C
B must be on AC
Segment: Part of a line
with end points
D
A
B
C
E
Name: DE; ED
Length:
 On a number line, it is the
absolute value of the
difference.
 The positive distance from
endpoint to endpoint
Opposite Ray:
 Has the same endpoint but goes
in the other direction
Postulates: accepted WITHOUT proof
Congruent: Two objects with the same size and shape
Congruent Segments: Segments with equal length
Midpoint of a Segment: Point that divides a segment into two congruent segments
Ray:

Has one endpoint and
goes for ever in one
direction
Acute angle: between 0 and 90
degrees
Right
exactly 90 degrees
Bisector of a Segment: A line segments, ray or plane that intersects with the segment
at angle:
a midpoint
Obtuse angle: between 90 and
190 degrees
Straight angle: exactly 180
degrees
Angle: An angle is formed by TWO rays what have the same endpoint
1.4 – Angles


The rays are the SIDES
Shared endpoint is the VERTEX
Angle addition postulate: If point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR.
Congruent angles: TWO angles with the SAME: size, shape, and measure.
Example:
m= the measure of an angle
m<R=M<S interchangeable  m<R=m<S
Adjacent angles: TWO angles that have a common vertex AND a common side, BUT no common interior
points.
ADJ angles:
Non ADJ angles:
5
7
6
8
Bisector of and angle: The rat that BISECTS and angle into two EQUAL adjacent angles
1.5 – Postulates and Theorems Relating Points, Lines, and Planes
POSTULATES:
Postulate 5:
 A line has at least two points
 A Plane has at least three points
 Space has at least four points
Postulate 6:
 Though any TWO points there is exactly one line
Postulate 7:
 F there are THREE points, at least one plane passes thought them
 If THREE points are non-collinear, exactly one plane passes thought them
Postulate 8:
 If TWO points are in a plane, then the line thought these points is also in the plane
Postulate 9:
 If TWO planes intersect then their intersection is ONE line
THEOREMS:
Theorem 1-1:
 If TWO lines intersect, then they intersect exactly one point.
Theorem 1-2:
 Thought a line and a point not on the line there is exactly one plane
Theorem 1-3:
 If TWO lines intersect, then exactly one plane contains the lines
Chapter 2: Deductive Reasoning
2.1- If-Then Statements; Converses
Conditional Statements:
 If-then statements are called conditional statements
 Can be TRUE or FALSE
Example: If today is Monday, then
 If: hypothesis, Then: conclusion
tomorrow is Tuesday. TRUE
Other forms of Conditional
Statements:
 If P then Q
 P implies Q
 Q if P
Converse:
 Switch P and Q ----to make----> If Q then P
 Can be TRUE or FALSE
Example:
Conditional Statement: If it’s raining,
then it’s cloudy. TRUE
Converse: If it’s cloudy, then it’s
raining. FALSE
Counterexample:
 Shows that a statement is false
 Satisfies the hypothesis only
 Only need one counterexample to prove that the statement is false.
Example: Is the statement true or false?
If B is on AC the B is between A and C.
Counterexample:
A
C
B
●-------------------●----------●
Bi-conditional:
If a conditional statement and its converse are both TRUE then they can be combined into ONE statement
using IFF (if and if).
Example:
 Statement: IF A=B then A≈B TRUE
 Converse: If A≈B then A=B TRUE
 Biconditional: If A=B IFF A≈B
2.3- Proving Theorems
Midpoint theorem:
If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB
Example:
A
M
B
●------------●------------●
Definition of Midpoint:
If M is the midpoint of AB then AM=MB
Angle Bisector theorem:
If ray BX is the bisector of <ABC then m<ABX = ½ of m(<ABC)
and m(<XBC)= ½ m(<ABC)
A
Example:
X
B
Reasons used in Proofs:
 Given information
 Definitions
 Postulates
 Properties from Algebra
 Theorems
C
Definition of Angle Bisector:
If ray BX is the bisector of <ABC, then m(<ABX) = m(<XBC) or <ABX ≈ <XBC
2.4 – Special Pairs of Angles
Complementary Angles:
2 angles whose measures add up to
90◦
Example: m<M + m<N = 90
 <M and <N are complements
 <M is a complement to <N
 <N is a complement to <M
Supplementary Angles:
2 angles whose measures add up to 180◦
Example: m<X + m<Y = 180
 <X and <Y are supplementary angles
 <X is a supplement to <Y
Vertical Angles:
When 2 lines intersect, the angles opposite each other are vertical
angles
Example:

 <1 and <3 are vertical
2
angles
 <2 and <4 are vertical
1
3
angles
4
Theorem:
Vertical angles are congruent
2.5-Perpendicular Lines
Perpendicular lines: Two lines that intersect to form right angles are perpendicular lines
Theorem: If two lines are perpendicular then they form congruent adjacent angles
Theorem: If two lines form congruent adjacent angles then the lines are perpendicular.
Theorem: If the exterior sides of two adjacent acute angles are perpendicular, then the angles are
complementary.
2.6-Planning a Proof
Theorem: If TWO angles are COMPLEMENTS of ≈ angles, then the TWO angles are ≈.
Theorem: If TWO angles are CUPPLEMENTS of ≈ angles, then the TWO angles are ≈.
Chapter 3
3.1 – Definitions
Non-intersecting lines: TWO lines that dose not intersect are either parallel or skew
Parallel lines: Coplanar lines that do not intersect
Skew lines: Non- Coplanar lines that do not interest.
Theorem: If TWO parallel planes are cut by a third plane; then the lines of intersection are parallel.
Transversal: A line that inter sets
TWO or MORE coplanar lines in
different points
Interior Angles: < 3, < 4, < 5, < 6
Exterior Angles: < 1, < 2, < 7, < 8
Alternate Interior Angles:
<4&<6
<3&<5
Same-Side Interior Angles:
<4&<6
<3&<5
Corresponding Angles:
< 1 & < 5, < 2 & < 6, < 8 & < 4, <
7&<3
t
1
3
2
j
4
5
7
6
8
k
3.2 – Properties of Parallel Lines
Postulate: If TWO parallel lines are cut by a transversal, then corresponding angles are congruent
Theorem: If TWO parallel lines are cut by a transversal, then alternate interior angles are congruent
Theorem: If TWO parallel lines are cut by a transversal, then same-side interior angles are supplementary.
Summary:
If TWO parallel lines are cut by a transversal…
 Corresponding angles are congruent
 Alternate-interior angles are congruent
 Same-side interior angles are supplementary
Theorem: If a transversal is perpendicular to one of the TWO parallel lines, then it is perpendicular to the
other line also.
3.3 – Properties of Parallel Lines
If two lines are cut by a transversal and
 Corresponding angles are congruent
 Alternate interior angles are congruent
 Same-side interior are supplementary
Then, the two lines are parallel
Theorem: Through a point not on a line, there is exactly ONE line parallel to the given line.
- - - - - - ●- - - - - - - -- - - -
Theorem: Though a point outside a line, there is exactly ONE line perpendicular to the given line
- - - - ●- - - - - - - - - - - - - - -
Theorem: If two lines are parallel to the third line, then all of the lines are parallel to each other
Theorem: Two lines, perpendicular to the same line are parallel.
Ways to prove two lines are parallel:
 Show that 2 corresponding angles are congruent.
 Show that 2 alternate interior angles are congruent.
 Show that 2 same-side interior angles are supplementary.
 Show that both lines are perpendicular to a third line
 Show that both lines are parallel to a third line.
3.4 – Angles of a Triangle
Definition: A triangle is the figure formed by 3 segments joining 3 NON-collinear points.
R
P
S
Vertex: each of the 3 points R, P, S
Sides: segments, PR, RS, SP
Angles : <P, <R, <S
Definition: Auxiliary- Line, Ray, Segment that may be added to a diagram to help a proof
Theorem: The sum of the measures of the angles of a triangle is 190◦
3.5-Angles of a Polygon
Polygon = many sides
Definition: A polygon is formed by coplanar segments (sides) such that
 Each segment intersects exactly TWO other segments one at each endpoint.
 No TWO segments with a common endpoint are collinear
Convex Polygon: No line containing a side of the polygon contains a point in the interior of the polygon
Diagonal of a polygon: segment joining TWO nonconsecutive vertices
Regular polygon: If a polygon is both equiangular and equilateral then it is a regular polygon
Theorem: The sum of the measure of the angles of a convex polygon with n sides is 180(n-2)
Theorem: The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex,
is 360◦
3.6 – Inductive Reasoning
Conclusion 1: On each of the first six days, Jim
attended his geometry class, Mrs. Lee, his
geometry teacher, gave a homework assignment.
Jim concludes that he will have geometry
homework every day he has geometry class.
Inductive Reasoning:
 Reach conclusions based on several past observations
 The conclusion is sometimes, but not always true
 Conclusion 1 above is an example of inductive reasoning.
Conclusion 2: In the same geometry class, Maria reads the theorem
“vertical angles are congruent.” She notices in a diagram that <1 and
<2 are vertical angles. Maria concludes tha <1 ≈ <2.
Deductive Reasoning:
 Reach conclusions based on accepted statements, including previous theorems, postulates,
definitions, and given information
 The conclusion must be true if the hypotheses are true.
 Conclusion 2 above is an example of deductive reasoning.
Chapter 4: Congruent Triangles
4.1 – Congruent Figures
Congruent Figures: Figures with the same shape and the same size are called ≈ figures.
Congruent Triangles: Two Triangles are ≈ iff their vertices can be matched up so that the corresponding
parts (angle & sides) of the Triangles are ≈.
Instead of “definition of ≈ Triangles”, we use “corresponding parts of ≈ triangles are ≈. (CPCTC)
4.2 – Some ways to Prove Triangles Congruent
SSS Postulate: If THREE sides of one Triangle are congruent to THREE sides of another Triangle, then
the Triangles are congruent.
SAS Postulate: If TWO sides and the included angles of one Triangle are congruent to TWO sides and the
included angle of another triangle, then the Triangles are congruent
ASA Postulate: If TWO angles and the included side of one Triangle are congruent to TWO angles and the
included side of another Triangle, then the triangles are congruent.
4.4 – The Isosceles Triangle Theorems
vertex
Isosceles Triangle:
leg
leg
 Legs = sides
 Base: 3rd side
 Base Angles: angle at base
 Vertex: angle opposite of the base
Theorem: If TWO sides of an Triangle are congruent, then the angles opposite
those sides are congruent.
base
Corollary: An equilateral Triangle is also equiangular.
Corollary: Bisector of the vertex angle of an isosceles Triangle is perpendicular
to the base at its midpoint
Theorem: If TWO angles of a Triangle are congruent, then the sides opposite those angles are congruent.
4.5 - Other Methods of Proving Triangles Congruent
Remember: SSS, ASA, SAS postulates
AAS Theorem: If TWO angles are ONE non-included side of ONE triangle is congruent to TWO angles
and ONE non-included side of another triangle, then the two triangles are congruent.
HL (Hypotenuse leg) Theorem: If the hypotenuse and a leg of a RIGHT triangle is congruent to the
corresponding parts of another RIGHT triangle, then the Triangles are congruent.
All Triangles: SSS, ASA, SAS, AAS
Right Triangle: HL
4.7 – Medians, altitudes, and Perpendicular Bisectors
Median: In a triangle is a segment from a vertex to the midpoint of the opposite side.
Altitude: An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the
opposite side.
Perpendicular Bisector: A perpendicular bisector of a segment is a line (or ray or segment) that is
perpendicular to the segment at its midpoint.
Distance from a point to a line: The distance from a point to a line (or plane) is defined to be the length of
the perpendicular segment from the point to the line (or plane).
Theorem: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the
endpoint to the segment.
Theorem: If a point is equidistant from the endpoint of a segment, then the point lies on the perpendicular
bisector of the segment.
Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the
angle.
Theorem: If a point is equidistant from the sides of an angle, then the point lies on the bisector of the
angle.
Chapter 5
5.1 – Properties of Parallelograms
Definition of a Parallelogram- a quadrilateral where both sides that are opposite sides are parallel.
Theorem: Opposite SIDES of a parallelogram are congruent.
Theorem: Opposite ANGLES of a parallelogram are congruent.
Theorem: Diagonals of a parallelogram bisect each other .
5.2 – Ways to prove that Quadrilaterals are Parallelograms
Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Theorem: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is
a parallelogram.
Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Theorem: If both of the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
Ways to prove that a quadrilateral is a parallelogram:
 Show that both pairs of opposite sides are parallel
 Show both pairs of opposite sides are congruent
 Show one pairs of opposite sides are both parallel and congruent
 Show both pairs of opposite angles are congruent
 Show the diagonals bisect each other
5.3 – Theorems Involving Parallel Lines
Theorem: If TWO lines are parallel, then all points on one line are equidistant from the other line
Theorem: If THREE parallel lines cut off congruent segments on one transversal, then they cut off
congruent segments.
Theorem: A line that contains the midpoint of one side of a triangle, and its parallel to another side, passes
thought the midpoint of the third side.
Theorem: The segment that joins the midpoints of TWO sides
 Parallel to the third side
 Parallel half as long as the third side
5.4 – Special Parallelograms
Definition of a Rectangle: A quad with four right angles. (Every rectangle is a parallelogram)
Theorem: The diagonals of a rectangle are congruent.
Theorem: If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
A Rectangle has (Properties):
 Four right angles
 Parallelogram
 Congruent Diagonals
Definition of a Rhombus: A quad with four congruent sides. (Every rhombus is a parallelogram)
Theorem: The diagonals of a rhombus are perpendicular
Theorem: Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem: If TWO consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
A Rhombus has (Properties):
 Four congruent sides
 Perpendicular diagonals
Definition of a Square: A quad with four right angles and four congruent sides. (every square is a
rectangle, rhombus and a parallelogram.)
A Square has:
 Four right angles
 Four congruent sides
 Diagonals are congruent and perpendicular
 Diagonals bisect its angles.
5.5. – Trapezoids
Definition: A quadrilateral with exactly one pair of parallel sides is a trapezoid
Definition: An Isosceles has congruent legs
Theorem: Base angles of an isosceles trapezoid are congruent
Definition: Median: segment that joins midpoints of legs
Theorem: The median of a trapezoid is:
 Parallel to the bases
 Has a length equal to the average of the bases lengths.
Chapter 6: Inequalities in Geometry
6.1- Inequalities
Properties of Inequality:
 If a>b and c>d, then a+c > b+d
 If a>b and c>0 then ac>bc and a/c > b/c
 If a>b and c<0 then ac<bc and a/c < b/c
 If a>b and b>c, then a>c
 If a = b +c and c> 0 then a>b
** In a proof: “A Prop. Or Inequality”
The Exterior angle Inequality Theorem: The measure of an ext. angle of a triangle is greater then the
measure of either remote interior angle.
6.4 – Inequalities for One Triangle
Theorem:
Theorem:
Corollary:
The Triangle Inequality Theorem:
Chapter 7: Similar Polygons
7.1 – Ratio and Proportion
Ratio: The ratio of one number to another is the quotient when the first number is divided by the second.
Ratio of 8 to 12 is 8/12 = 2/3
If y ≠ 0 the ratio of x to y is x/7 OR x :y
Proportion: A proportion is an equation stating that two ratios are equal.
a/b = c/d or a :b = c :d
Extended proportion: a/b = c/d = e/d
7.2 – Properties of Proportions
a/b = c/d
Means: middle terms (b, c)
Extremes: first and last (a, d)
Means- Extremes Property: The products of the extremes equal the product of the means.
Properties of Proportions: In a proof our reason will state “ A Prop. Of Proportions”
a/b = c/d
Equivalent to:




ad = bc
b/a = d/c
a/c = b/d
a +b/ b = c +d/d
7.3 – Similar Polygons
Similar Triangles: Two polygons are similar if their vertices can be paired so that
1. Corresponding angles are congruent
2. Corresponding sides are in proportion (lengths have the same ratio)
7.4 – A Postulate for Similar Triangles
AA Similarity Postulate: If TWO angles of one triangle are congruent to TWO angles of another triangle,
then the triangles are similar.
7.5 – Theorems for Similar Triangles
SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the
sides including those angles are in proportion, then the triangles are similar.
SSS Similarity Theorem: If the sides of two triangles are in proportion, then the triangles are similar
7.6 – Proportional Lengths
Definition: Point L and M lie on AB and CD respectively. If AL/LB = CM/MD then AB and CD are
divided proportionally.
Triangles Proportionality Theorem: If a line parallel to one side of triangle intersects the other two sides,
then it divides those sides proportionally.
Corollary: If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Triangles Angle-Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite sides
into segments proportional to the other two sides.
Common answers to Proofs
Properties of Equality:
 Addition POE:
1. If [A=B] and [C=D] then [A+C= B+D]
 Subtraction POE:
1. If [5 = 5], then [5 - 2 = 5 – 2]
 Multiplications POE:
1. [A=B] then [AC=AB]
 Division POE:
1. If [A=B] and [C≠0] then [A/C = B/C]
 Substitution POE:
1. If [m(<1)=m(<2)] and [m(<3)=m(<2)] then [m(<1)=m(<3)]
 Reflexive Property:
1. [a=a]
2. [BC=BC]
 Symmetric Property:
1. If [a=b] then [b=a]
 Transitive Property:
1. If [a=b] and [b=c] then [a=c]

Properties of Congruence:
Symmetric Property:


1. If [line AB] is congruent to [line XY] then, [line XY] is congruent to [line A]
Reflexive Property:
1. [<A] is congruent to [<A]
Transitive Property:
1.
If [<A is congruent to <B] and [<B is congruent to <C] then [<A is congruent to <C]

Definitions:
Def. or Angle Bisector: If [line BX] is the bisector of <ABC then m(<ABX) = m(<XBC) or m(<ABX)
≈ m(<XBC)
Def. of Complementary angles: two angles whose measure add up to 90º

Def. of Supplementary angles: two angles whose measures add up to 180 º

Def. of Midpoint: If M is the midpoint of [AB] then [AM=MB]













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Theorems:
If two lines that intersect that form right angles then they are perpendicular lines.
If two angles are complements of congruent angles then the two angles are congruent.
If two angles are supplements of congruent angles then the two angles are congruent.
Angle bisector theorem: If [line BX] is the bisector of [<ABC] then [m<ABX=1/2>ABC] and
[m<XBC=1/2<ABC]
Midpoint theorem: If M is the midpoint of [AB] then [AM=1/2 of AB]
Vertical angles: all vertical angles are congruent.
If two lines are perpendicular then they form congruent adjacent angles.
If two lines form congruent adjacent angles then the lines are perpendicular.
If the exterior sides of two adjacent acute angles are perpendicular then the angles are complementary.
If two parallel planes are cut by a third plane; then the lines of intersection are parallel.
If two parallel lines are transversal; then alternate interior angles are congruent.
If two parallel lines are cut by a transversal then same side interior angles are supplementary.
Through a point not on a line, there is exactly one line parallel to the given line.
Though a point outside a line, there is exactly one line perpendicular to the given line.
Though a point outside a line, there is exactly one line perpendicular to the given line.
If two lines are parallel to a third line. They are parallel to each other (J//K, K//L then J//L)
Two lines perpendicular to the same line are parallel (If J is perpendicular to L and K is perpendicular
to L then J//K)
Postulates:
Segment Addition Post:
1. If XY + YZ = XZ then point Y is between points X and Z
2. If point Y is between points X and Z then XY + YZ = XZ
Angle Addition Post:
1. If point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR.
If two parallel lines are cut by a transversal then the corresponding angles are congruent Post.
Ways to prove two lines are parallel:
 Show that 2 corresponding angles are congruent.
 Show that 2 alternate interior angles are congruent.
 Show that 2 same-side interior angles are supplementary.
 Show that both lines are perpendicular to a third line
 Show that both lines are parallel to a third line.
CPCTC:
SSS Postulate: If THREE sides of one Triangle are congruent to THREE sides of another Triangle, then the
Triangles are congruent.
SAS Postulate: If TWO sides and the included angles of one Triangle are congruent to TWO sides and the
included angle of another triangle, then the Triangles are congruent
ASA Postulate: If TWO angles and the included side of one Triangle are congruent to TWO angles and the
included side of another Triangle, then the triangles are congruent.