Alex Weinberg
Math Midterm Review
January 2011
To find out if two statements are logically equivalent we use
truth tables.
If you have more than 2 variables than to find the number of
rows you need, you have 2x with (x=number of variables)
Law of Detachment: If → then and the 'if' = true
You can conclude that the 'then' = true
p→q
p
q
Law of Contrapositive: In an implication you are allowed to
conclude it's contrapositive
p →q
q → p
Modus Tollens: In an implication if you have negation of 'then' you may
conclude the negation of the 'if'
p→q
q
p
Chain Rule
p→q
q→r
p→r
Disjunctive Inference: In an 'or' statement if one of the things
is false then the over one must be true.
pq
q
p
DeMorgan's Law:
pq)= pq
Law of Simplification
pq
p
q
Law of Conjunction
p
q
pq
Disjunctive Addition
p
pq
In an Indirect Proof assume the thing you want to prove is
false and then continue on with the proof from there.
Quantifiers
= all
= some
(b) = b)
bb)
Undefined terms – point, line, plane
Defined terms- words based on definitions
Line Segment – If you have two points on a line then the points between them form
a line segment
Intersection – the overlap between two segments/angles
Union- Everything together
Ray- if ray then starts at one point on a line and continues out in one direction
Angle If two rays share an endpoint then they form an angle
Degree- measurement of an<
Minute- 1/60 of degree- 60 minutes=degree
Second- 1/60 of minute- 60 seconds=minute
-every number is 30°
Congruent – Segments or Angles that have exact same measurement
Collinear- 3 points that are in a straight line
Non-Collinear- 3 points that are not in straight line
-2 points always collinear
triangle=3 nonlinear points connected
Triangle Rule: The sum of any two sides of a triangle must be greater than the 3rd
Allowed to Assume
Not Allowed to assume
Straight lines
Straight <
Supplementary
(acute and obtuse is argued about)
right<
measurements
congruence
perpendicular
complementary <
Theorems:
1.
2.
3.
4.
5.
6.
7.
If two angles are right angles then they are congruent
If midpoint/bisector/trisector then ≅ parts
If two lines perpendicular then they for right angle
If the sum of 2<s is 90° then they are complementary
If the sum of 2<s is 180° then they are supplementary
If 2<s supp/comp to same < then they are ≅
If 2≅ <s are added to ≅ angles then the whole angles are ≅ (addition
property pt. 1)
If two ≅ <s are added to same then the 2 whole angles are ≅ (part.2)
If 2 ≅ <s are subtracted from ≅ s then the angles are ≅ (≅ wholes-≅
parts= ≅ parts)(subtraction property part 1)
8. If 2 ≅ <s are subtracted from same then the result is≅ (Part 2)
If ≅ angles are divided by bisector etc. then the wholes are ≅
(Multiplication Property)(≅ <s bi then wholes≅)
9. If ≅ wholes divided by bisector etc. then the resulting parts are≅
(Division Property)
10. If 2<s are ≅ to ≅ <s then they are congruent
11. If 2<s are ≅ to same <s then they are ≅
Transitive
12. n/a
Property
13. Substitution property- If 2 <s are≅ then they
are interchangeable
14. If vertical <s then they are ≅
Vertical <s- A pair of angles that are across from each other when lines intersect
Bisector-Divides an angles or segment into two equal parts
Midpoint-Divides a segment into two congruent parts
Trisector-Divides an angle of segment into three congruent parts
Probability
Number of winners
Number of possibilities
Perpendicular-If two lines are perpendicular then they form a right angle
Congruent Triangles
Triangle- is the union of 3 non-collinear points
Congruent Triangles- if you put one triangle on top of the other they would match
exactly
If 2 triangles are ≅ then the corresponding parts are ≅
Ways to prove triangles congruent
1. 3 ≅ sides and angles
2. SSS
3. SAS
4. ASA
5. AAS
CPCTC
Corresponding Parts Congruent Triangles Congruent
After you prove that 2 triangles are congruent you can prove that any corresponding
part is congruent
Circle- set of points that are equidistant from one point, called the center point
Radius- A segment connects the center point to a point on the circle
If radii then congruent
Auxiliary Line- If you have 2 points anywhere then you can connect them into a
line
If 2 <’s are supplementary and congruent then they are both right angles
Median of a Triangle- segment that goes from a vertex of a triangle to midpoint of
opposite side
Altitude of a Triangle- segment that goes from vertex angle to the opposite side,
forming a right angle
Types of Triangles
Angles
Sides
1. Acute- All 3 angles are acute
1. Equilateral- all sides congruent
2. Obtuse- 1 angle obtuse and other
2. Isosceles- at least 2 sides
two angles are acute
congruent
3. Right - one angle is 90
3. Scalene- all sides are different
4. Equiangular- all angles are
congruent
Isosceles Theorem- If legs ≅ then base angles ≅ (reverse is also true)
Hypotenuse Leg- basically ASS as long as the A is the right angles in a right triangle
Detour Proof- prove one set of triangles congruent so that you can prove another
set is too
Case of the Missing Diagram
1. Draw a diagram
2. List your givens
3. State the thing
E1- If two points are equidistant from the endpoints of a segment (congruent
segments) then they form the perpendicular bisector of that segment
E2- if a point is on a perpendicular bisector of a segment then it is equidistant from
the endpoints of that segment
E1
Proving a perpendicular bisector
E2
Proving a point is equidistant from 2
points
Lies On – If a point equidistant from endpoints of a segment, then it must lie on the
perpendicular bisector of the segments
Biggest side- angle across from it is biggest
Same for smallest side
Plane- Many lines next to each other infinitely
Coplanar- on the same plane
Non-Coplanar- on different plane
Parallel Lines- coplanar lines that never intersect
Skew Lines- non-coplanar lines that never intersect
Transversal- line that intersects 2 or more lines (in one place)
Parallel Theorems
1. Alt int ≅
2. Alt ext ≅
3. Corresponding ≅
4. Consec int supp
5. Consec ext sup
Parallel Postulate- you are allowed to draw a line and say it is parallel to another
line
Exterior Angle Theorem- exterior angle is equal to the other two angles combined
Midline Theorem- a. midsegment is always parallel to the third side
b. length of midsegment is equal to ½ of the third side
Proving Line Parallel
1. reverse the properties
2. 2 line perpendicular to the same line
No Choice- if 2 angles of triangles are congruent then you can conclude that the 3rd
angle is as well
Sum of Interior angles- 180(n-2)
Number of Diagonals- n(n-3)
Sum of Exterior angles- 360
Properties of Shapes
Parallelogram
1. opposite sides parallel
2. opposite angels congruent
3. opposite sides congruent
4. consecutive angles sup
5. diagonals bisect each other
Rectangle
1. 5 parallelograms properties
2. diagonals congruent
3. all four angles are right
Rhombus
1.
2.
3.
4.
5 parallelogram properties
all sides congruent
diagonals bisect angles
diagonals are perpendicular bisectors to each other
Square
1. 5 properties of parallelogram
2. 2 properties of rectangle
3. 3 properties of rhombus
Kite
1.
2.
3.
4.
2 different pairs of consecutive sides congruent
longer diagonal perpendicular bisector of shorter diagonal
opposite angles of the shorter diagonal are congruent
longer diagonal bisects the angles
Trapezoid
1. Pair of parallel sides
2. Pair of non-parallel sides
3. Random upper angle is sup to any lower angle
Iso Trapezoid
1. non-parallel sides congruent
2. lower angles congruent
3. upper angles congruent
4. diagonals congruent
Ways to prove
Parallelogram
1. Opposite sides parallel
2. Both pairs of opposite sides congruent
3. Diagonals bisect each other
4. Both pairs opposite angles congruent
5. Consecutive angles supplementary
6. One pair of opposite sides congruent and parallel
Rectangle
1. Parallelogram + one right angle
2. All angles right
3. Parallelogram + diagonals congruent
Trapezoid
1. Quadrilateral w/one pair of parallel sides
a. Must show other pair non-parallel
Iso Trapezoid
1. Trap + non-parallel sides congruent
2. Trap + upper base angles congruent
3. Trap + lower base angles congruent
4. Trap + diagonals congruent
Rhombus
1. Parallelogram + consecutive sides
2. Diagonals perpendicular bisectors of each other
3. Parallelogram + diagonals bisect angles
4. All four sides congruent
Square
1. Rectangle + Rhombus