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Mathematical Sentence - a sentence that states a fact or complete idea
Open sentence – contains a variable
Closed sentence – can be judged either true or false
Truth value – true/false
Negation – “not” (~)
* Statement and its negation have opposite truth values*
Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)
Disjunction is true when either one part or both parts of the statement is true. (P is true, q is false
P\/q is true)
Give the original statement p  q
Converse is formed by interchanging the hypothesis and conclusion. q  p
Inverse is formed by negating the hypothesis and conclusion. ~p  ~q
Contrapositive is formed by negating and switching the hypothesis and conclusion. ~q  ~P
*The conditional and its contrapositive are logically equivalent*
Biconditional – conjunction of a conditional and its converse (p q) ^ (q p)
Pq means (p q) ^ (q  p)
It is true when both p and q have the same truth value.
*Tautology is a compound statement that is always true. Opposite of a Tautology is a Contradiction.*
Ex.
P
q
[p^(~p\/q)] (p^q)
T
T
T
T
F
T
F
T
T
F
F
T
Contradiction – a compound statement that is always false
Law of Detachment
P q (true)
p
(true)
:. q is also true (conclusion)
Law of Modus Tollens
P  q (true)
~q
(true)
:. ~p (true) OR [(p à q) ^ ~q] à ~P
Law of Syllogism
p q
q r
:. p r
Law of Detachment (Modus Ponens)
p q
p
:. q
Law of Modus Tollens
p q
~q
:. ~p
Line – a set of points that extend indefinitely in both directions.
Plane – a set of points which form a flat surface and extend indefinitely in all directions.
Collinear points – points that are on the same line.
Coplanar points – points that are on the same plane.
Line segment or segment – a set of two points called endpoints and all the points between them.
Ray – part of a line that starts at one point (called the endpoint) and extends endlessly in one direction
Basic Properties/Postulates
Reflexive Property – a quantity is equal to itself
Symmetric Property – an equality may be expressed in any other order
Transitive Property – if quantities are equal to the same quantity, then they are equal to each other
Substitution Property – a quantity may be substituted for its equal in any expression
Midpoint – a midpoint divides a segment into 2 congruent segments
How to Interpret a Diagram
Can Assume
Partition Postulate – the whole is equal to the sum of its parts
Median of a triangle – a segment drawn from ant vertex of a
triangle to the MIDPOINT of the opposite side
Altitude of a triangle – a segment drawn from any vertex of the
triangle perpendicular to the line containing the opposite side
Circle – a set of points in a plane that are a given distance from a
given point in that plane
1. Straight lines
and angles.
2. Collinearity of
points.
3. Between-ness
of points.
4. Relative
position of
points
Radius – a segment drawn from the center of a circle to any point
on the circle
ALWAYS start off with the Postulate of the Excluded Middle in an indirect Proof
Distance between two points is the length of the line segment connecting two points.
●
●
Can’t Assume
1. Right angles
2. Congruent
segments.
3. Congruent
angles.
4. Relative sizes
of segments
and angles
(not drawn to
scale)
Distance from a point to a line is the length of the perpendicular segment from the point to the line.
●
●
●
The triangle Inequality Postulate - In a triangle, the sum of two side lengths is greater than the length of
the third side.
Parallelogram – a quadrilateral in which both pairs of opposite sides are parallel
Each diagonal of a parallelogram splits it into two congruent triangles.
Properties:
1.
2.
3.
4.
Both pairs of opposite sides are congruent.
Both pairs of opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Proving a Quadrilateral is a parallelogram
1.
2.
3.
4.
5.
6.
Show that both pairs of opposite sides are parallel
Show that one pair of opposite sides are both parallel and congruent
Show that both pairs of opposite sides are congruent
Show that both pairs of opposite angles are congruent
Show that diagonals bisect each other
An angle is supplementary to both of its consecutive angles
Rectangle – a parallelogram in which at least one angle is a right angle.
Properties
1. Equiangular
2. All the properties of a parallelogram
3. Diagonals are congruent
Proving a Quadrilateral is a Rectangle
1. A parallelogram with at least one right angle
2. A parallelogram with congruent diagonals
3. Equiangular
Rhombus – A parallelogram in which at least two consecutive sides are congruent
Properties
1.
2.
3.
4.
Equilateral
All the properties of a parallelogram
Diagonals are perpendicular
Each diagonal bisects a pair of opposite angles
Proving a Quadrilateral is a Rhombus
1.
2.
3.
4.
A parallelogram with at least two adjacent sides congruent.
A parallelogram whose diagonals are perpendicular
A parallelogram whose diagonals bisect one angle of the parallelogram
Equilateral
Square – a parallelogram that is both a rhombus and a rectangle
Properties
1.
2.
3.
4.
5.
6.
Equiangular
All the properties of a parallelogram
Diagonals are congruent
Equilateral
Diagonals are perpendicular
Each diagonal bisects a pair of opposite angles
Proving a Quadrilateral is a Square
1. A rhombus with one right angle
2. A rectangle with two congruent adjacent sides
Trapezoid – a quadrilateral with exactly one pair of parallel sides
Isosceles trapezoid – a trapezoid in which the nonparallel sides are congruent
Properties
1. Lower base angles are congruent
2. Upper base angles are congruent
3. Diagonals are congruent
Kite – a quadrilateral with two disjoint pairs of consecutive sides are congruent
Properties
1. One diagonal is the perpendicular bisector of the other.
2. One of the diagonals bisects a pair of opposite angles.
3. One pair of opposite angles are congruent.
Ratio – a quotient of two numbers
Proportion – an equation stating that two or more ratios are equal
Mean Proportion – a proportion in which the means are equal
Ex. ½ = 2/4 = 3/6
Triangle Mid-segment Theorem - A segment joining the midpoints of two sides of a triangle is parallel to
the third side and its length is ½ the length of the third side.
Median (mid-segment) of a trapezoid – the segment joining the midpoints of the nonparallel sides of a
trapezoid
Similar polygons (~Polygons) – polygons in which
1. All pairs of corresponding angles are congruent
2. The ratios of the lengths of all pairs of corresponding sides are equal
Proving triangles similar – if there exists a correspondence between the vertices of two triangles
such that
(AA~ Theorem) Two angles of one triangle are congruent to the corresponding angles of the other then
the triangles are similar.
(SSS~ Theorem) the ratio of the measures of the corresponding sides are equal then the triangles are
similar.
(SAS~ Theorem) the ratios of the measures of two pairs of corresponding sides are equal and the
included angles are congruent, then the triangles are similar.
Side Splitter Theorem (Triangle Proportionality Theorem) – if a line is parallel to one side of a triangle
and intersects the other two sides, it divides those sides proportionally
Theorem – if an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed
are similar to the given right triangle and to each other
Leg 1
Segment 1
Altitude
Leg 2
Segment 2
Segment 1 = Altitude
Altitude
Segment 2
Hypotenuse = Leg 1
Leg 1
Segment 1
Altitude Rule – The altitude to the hypotenuse of a right triangle is the mean proportional between the
segments into which it divides the hypotenuse.
Leg Rule – Each leg of a right triangle is the mean proportional between the hypotenuse and the
projection of the leg of the hypotenuse.
Angle Side Measures
30o – 60o – 90o
60o
2x
45o – 45o – 90o
45o
X
x
90o
x√2
30o
x√3
90o
45o
x
Areas
Area of an Equilateral Triangle = (Side)2√3/4
Area of a Rhombus = d1d2/2
d2
S
d1
Scale factor = Ratio of Similitude  s1/s2 = P1/P2
(s stands for Side, P stands for Perimeter)
Area of a Square = (Side)2
Area of a Rectangle = (Base)(Height) OR (Length)(Width)
Area of a Triangle = ½(Base)(Height)
Area of a Trapezoid = ½ (Base 1+Base 2)(Height) OR (Median)(Height)
Area of a N-gon = ½ (Apothem)(Perimeter)
Median Of a Trapezoid = (Base1+Base2)/2
Radius of a regular polygon is a segment joining the center to any vertex
Apothem of a regular polygon is a segment joining the center to the midpoint of any side
Lines
Distance Formula
Distance = √(Δx)2+(Δy)2
Midpoint Formula
MAB = (xA +XB/2),(YA+YB/2)
The midpoint is the Average of the two points you’re finding the midpoint for.
Slope is the rate of change in y-value for each unit of change in x-value.
Parallel lines have equal slopes.
Perpendicular lines have slopes whose product is -1. (Their slopes are negative reciprocals of each
other.)
Of parallel lines share a point, they are non-distinct.
Ways to Write a Line Eqauation
Point Slope Formula
1. ax+by=c
2. y-y1=x-x1(m)
3. y=mx+b
Ways to Solve Linear Equations
1. Graphical
(x,y)
3x+5y=10
Line b
x-7y=12
Line a
2. Substitution
Given equations
3x+5y=10
X=12+7y
Find point of intersection of these two lines.
3(12+7y)+5y=10
36+21y+5y=10
26y=-26
Y=-1
3. Elimination
10x+3y=2
6x-7y=-12
x=12+7y
x=12+7(-1)
x=5
 70x+21y=14
 +18x-21y=-36
88x
= -22
88
88
X=-1/4
Point of intersection: (5,-1)
10x+3y=2
10(-1/4) +3y=2
-5/2+3y=2
3y=9/2
y=3/2
Point of Intersection: (-1/4,3/2)
Quadratic Equations
Quadratic equations can be written in the form y=ax2+bx+c
Axis of Symmetry – Vertical Line given by x OR x= -b/2a
Vertex; min; max; turning point
Vertex found by Substitution of x=-b/2a into quadratic equation.
To find the y intercept, set x=0
Steps for Graphing Parabolas
1. Find Axis of Symmetry (x=-b/2a)
2. Use #1 to find Vertex.
3. Plot 2-3 points on one side of Axis of Symmetry. (y intercept is an easy one, it’s simply c in
y=ax2+bx+c)
4. Reflect each point across axis of symmetry.
5. Sketch an expanding u-shape with arrows.
Circle Equation
R2=(x-h)2+(y-k)2
At a glance, you can see that (h, k) is the center of the circle.
This is in Center Radius form.
Completing the Square
To complete the square,
halve the linear coefficient
for x and y, then add its
square to both sides of the
equation.
Ex.
x2+y2+6x-8y-24 = 0
x2+6x+___+y2-8y+___=24
(x+3)(x+3)
(y-4)(y+4)
x2+6x+9+y2-8y+16=24+9+16
(x+3)2 + (y-4)2 = 49
Center of the circle = (-3, 4)
Radius = 7
These satisfy the given equation.
Graphs
Solve algebraically, check graphically.
3 Scenarios of Graphs
1. One intersection
2. Two intersections
3. Nothing, no slope
Circular Geometry
Two or more coplanar circles with the same center are called concentric circles.
Two circles are congruent/ equal if they have congruent/equal radii.
A point is in the interior of a circle if its distance from the center is less than the radius.
A point is in the exterior of a circle if its distance from the center is larger than the radius.
A point is on a circle if its distance from the center is equal to the radius.
A chord of a circle is a segment joining any 2 points on the circle.
A diameter of a circle is a segment (chord) that passes through the center of the circle.
The distance from the center of a circle to a chord is the measure of the perpendicular segment from
the center to the chord.
The circumference of a circle is its perimeter.
Sector of a circle is a region bounded by two radii and their intercepted arc denoted by 3 letters: center
and two endpoints of the arc.
Locus
Locus a set of points satisfying a given equation
Usual Loci
a. All points equidistant from a fixed distant, d
It is a circle with center d.
b. All points equidistant from 2 points A&B
The perpendicular bisector of line AB
c. All points fixed distance d, from line l
Two lines, each parallel to line l , d units to either side of line l
d. All points equidistant from 2 parallel lines m & n
One line parallel to m & n equidistant from each line
e. All points equidistant from 2 intersecting lines.
Two lines bisecting angles formed by two given lines
f. All points equidistant from sides of an angle.
An angle bisector of that angle
The perpendicular bisectors of a triangle are concurrent at a point equidistant from the vertices. This
point is called the circumcenter.
The angle bisectors of a triangle are concurrent at a point equidistant from the sides of the triangle. This
point is the incenter.
The lines containing the Altitudes of a triangle are concurrent at a point called the orthocenter.
The medians of a triangle are concurrent at a point 2/3 of the way from any vertex to its opposite sides.
This point is the centroid.
Line Symmetry
A figure has line symmetry if a line can be drawn such that each side is a mirror image of the other.
Transformation – change in position, size, or orientation of a figure
Line Reflection - transformation that produces a mirror image of a figure on opposite side of the given
line
r y-axis (x, y) = (-x, y)
r x-axis (x, y) = (x, -y)
r y=x (x, y) = (y, x)
r y=c (x, y) = (x, 2c-y)
r x=c (x, y) = (2c-x, y)
R o (x, y) = (-x, -y)
A figure has point symmetry if it is its own image under a reflection in a point.
R o, 90 (x, y) = (y, -x)
R o, 180 (x, y) = (-x, -y)
R o, 270 (x, y) = (y, x)
R y=-x (x, y) = (-y, -x)
T 3, -2 (x, y) = (x+3, y-2)
If you can rotate your
figure 180o about the
point in question, and it
is still what it used to
be, then it has point
symmetry.
D k (x, y) = (kx, ky)
Isometry a transformation that preserves distance
Direct isometry is one that preserves order (orientation)
Opposite isometry is one that changes order, or orientation, from clockwise to counterclockwise, or
vice-versa.
To reflect an
equation in y=x,
switch x and y then
solve for y.
Space Geometry
Which of these determine a PLANE ?
1 point
No
Two points
No
Three collinear
points
No
Three noncollinear points
Yes
A line and a
point not on the
line
Yes
Two intersecting
lines
Yes
Two parallel lines Yes
Lateral Area & Total Area
Lateral Area of a Cylinder = 2πrh
Total Area of a Cylinder = 2πrh + 2πr2
Lateral Area of a Cone = πrl
Slant Height (l )
Total Area of a Cone = πrl + πr2
Total Area of a Sphere = 4πr2
Volume
Volume of a Cylinder = πr2h
Volume of a Prism = l●w●h
Volume of a Cone = 1/3 πr2h
Volume of a Regular Pyramid = 1/3 (Area of the base)(height)
Altitude and Slant
Height are
different…Altitude
goes from the tip
of the cone to the
bottom.
Volume of a Sphere = 4/3 πr3
Sites you’d want to visit.
http://www.regentsprep.org/regents/math/geometry/GPB/theorems.htm
www.jmap.org
You should also check out the back of your text book, it has all the theorems/postulates. C:
GOOD LUCK ON THE REGENTS!!
---Navida Rukhsha c: