Properties of Equality
Geometric Examples
Addition If a = b If AB = CD
Property: then a + c = b + c then AB + BC = CD + BC
SubtractionIf a = b If AC = BD
Property: then a -- c = b – c then AC -- BC = BD -- BC
Multiplication If a = b If AB = 1/3 AC Property: then a x c = b x c then 3 (AB) = 3 (1/3 AC)
( ac = bc ) then 3 (AB) = AC
Division If a = b If 3 (AB) = AC
Property: then a ÷ c = b ÷ c then 3 (AB) = (AC) ( a = b ) 3 3
c c then AB = 1/3 AC
Substitution If a = b If AB + BC = AC Property: Then b can substitute and QS = AC
for a in any equation. then AB + BC = QS
Reflexive a = a AD = AD
Property:
Symmetric If a = b If AD = QT
Property: Then b = a then QT = AD
Transitive If a = b If AB = CD
Property: And b = c and CD = ST
Then a = c then AB = ST
Definitions:
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Definition of Congruence:
Exactly equal in size and shape Definition of Congruent Angles:
Two angles that are exactly equal in size and shape Definition of Congruent segments:
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Two line segments that are exactly equal in size and shape Definition of a bisector
Cuts a segment or geometric shape exactly in half (2 equal parts) Definition of Angle Bisectors
Cuts an angle into two equal angles Definition of a Line
The shortest distance between two points; continues to infinity Definition of a midpoint:
The exact middle of a line segment; creates two congruent segments Definition of Complementary Angles
Two angles that add up to 90° Definition of a Perpendicular Angle
One angle that is exactly 90° Definition of Supplementary Angles
Two angles that add up to 180° Definition of a Linear Pair
Two angles that create a straight line (supplementary); add up to 180°; or a straight line that is divided into two angles
Postulate -- a statement that is assumed true without proof
Segment Addition Postulate:
2 smaller segments add up to the combined longer segment
AB + BC = AC AB + BC + CD = AD
Angle Addition Postulate:
2 smaller angles add up to the combined inclusive larger angle
m<ASB + m<BSC = m<ASC
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Theorem – a true statement that can be proven
Common Segment Theorem:
If two equal segments share the same segment, then the combined larger segments are also equal.
AB = CD
BC = BC
AB + BC = CD + BC AB + BC = AC
CD + BC = BD AC = BD
Vertical Angle Theorem:
Vertical angles are always congruent (equal)
m<ASC = m<DSB
m<ASD = m<CSB
Linear Pair Theorem:
If two angles form a linear pair (combine to make a straight line) then they are supplementary (add up to 180°).
If AB = a straight line, then m<ASC + m<CSB = 180°
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LOGIC:
Statement–a declarative statement that is either true or false, but not both. It cannot be a question or an imperative command. It can be a regular statement or an if/then conditional statement.
Ex. YES: It’s cold outside today. If you put on a sweater, you will be cold. NOT: Go to bed. Where is my turtle?
Conditional Statements—statements that use a hypothesis and deductive reasoning to draw a conclusion; usually represented by If…then…
IF _________________________, THEN ______________________________
In logic notation, the hypothesis = p and the conclusion = q Therefore the condition statement can also be written as
p
q
Converse Statement—the opposite of your first conditional statement, obtained by switching the hypothesis and the conclusion
IF _________________________, THEN ______________________________
q
p
Inverse Statements—the same as the conditional, except with “not” in front of both the hypothesis and conclusion
A B
S
HYPOTHESIS CONCLUSION
HYPOTHESIS
CONCLUSION
IF _________________________, THEN ______________________________
~
p
~q
Contrapositive Statement— the same as the converse, except with “not” in front of both the hypothesis and conclusion
IF _________________________, THEN ______________________________
~q
~
p
Counterexample—any example that shows the statement is false.
The converse of a true conditional is often NOT true.
Ex. Conditional: If Ralph is a house cat, then he is a mammal. Converse: If Ralph is a mammal, then he is a house cat Counterexample: A dog is a mammal.
The conditional is true. The converse is false because there are many counterexamples of mammals that are not house cats.
Euler Diagram: visual representation of conditional statements
House cats are included in the whole of mammals.
Dogs and horses are also mammals, but they are not cats.
Biconditional—when both the conditional and its converse are true, they are combined into this one statement using “if and only if”
EX.
IF
p
q = true
AND
q
p = true
THEN
p
if and only if
q
EX.
IfC is on the perpendicular bisector of AB, then AC = CB. (true)
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(Counterexamples)
dogs horses
q
(conclusion)
mammals
p
(hypothesis)
house cats
NOT HYPOTHESIS
If AC = CB, thenC is on the perpendicular bisector of AB. (true) Therefore, C is on the perpendicular bisector of AB if and only if AC = CB.
Law of Detachment—If the hypothesis of a true conditional is true, then the conclusion is also true.
(separate conditional into two separate sentences)
EX.
Ifa figure is a rhombus, then it has 4 sides. (true)
Detachment--A figure is a rhombus. (true) It has 4 sides. (true)
Syllogism—If the first conditional statement is true. Take the conclusion from the first conditional and make it the hypothesis of the second conditional. If both conditionals are true, then the first hypothesis equals the second conclusion. If A = B, and B = C, then A = C
EX.
Ifa figure is a rhombus, then it has 4 sides. (true)
If a figure has 4 sides, then it is a quadrilateral. (true)
Syllogism-- If a figure is a rhombus, then it is a quadrilateral. (true)
Truth value—whether a statement is true or false
Negation—adding “not” to a statement to create its inverse
Contradiction—a situation in which a statement and its negation have the same truth value
Ex. Given: AC = CB
AC = CB Negation: AC ≠ CB
Truth value: true false
Therefore, it is not a contradiction (different truth values)
I always lie. I do not always lie.
Truth value: (false) (false)
If you admit you are lying, then you are telling the truth, so you are not lying about it.
If you said you always lie, then you cannot tell the truth. Therefore, the statement, “I always lie” is a contradiction.
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Ex. Given: Quadrilateral ABCD with m<A = 80° Prove: ABCD is not a rectangle
(Negation: ABCD is a rectangle) Indirect proof: Assume ABCD is a rectangle.
If ABCD is a rectangle, then all four angles are right angles But using the given, m<A = 80°
Since at least one angle is not 90°, we know that
the assumption that ABCD is a rectangle is false. Therefore, ABCD is not a rectangle.
Truth Tables
Conjunction “and”—both have to be true for it to be true
Disjunction “or”—only one has to be true for it to be true
Implication—If you are telling the truth about the hypothesis, then the conclusion is true; it is only false if your conclusion is false
Equivalent—the double implication is true if both are true or both are false; otherwise, it is false if you have one of each
Conjunction Disjunction Implication Equivalent
P Q P^Q P Q PvQ P Q P→Q P Q P↔Q
T T T T T T T T T T T T
T F F T F T T F F T F F
F T F F T T F T T F T F
F F F F F F F F T F F T
LINES AND TRANSVERSALS
Parallel lines—lines that lie in the same plane and never touch; they are also equidistant from each other and have the same slope
Transversal—a line that intersects two or more lines in a plane
Corresponding angles—angles that are in the same position on matching lines— same side of transversal, but one is inside & one is outside (top left, top right, bottom left, bottom right)
(1 = 5 2 = 6 3 = 7 4 = 8 )
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Alternate Interior or Exterior Angles—two nonadjacent interior or exterior angles that lie on opposite sides of the transversal (3 = 6 4 = 5)
(1 = 8 2 = 7)
Same-side Interior or Exterior Angles—two interior or exterior angles that lie between or outside the lines and are on the same side of the transversal
( 3 & 5 4 & 6 1 & 7 2 & 8 )
Angles that are Congruent Angles that are Supplementary
Corresponding Linear Pair
Alternate Interior Same-Side Interior Alternate Exterior Same-Side Exterior Vertical Angles
1 = 4 = 5 = 8 1 + 2 = 180° 3 + 5 = 180 °
PARALLEL LINE/ANGLE RELATIONSHIPS
Conjecture: If two parallel lines are cut by a transversal, then the corresponding angles are congruent
Converse: If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel.
Conjecture: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent
Converse: If two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.
Conjecture: If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary.
Converse: If two lines are cut by a transversal and the consecutive interior angles are supplementary, then the two lines are parallel.
SLOPES
For any two points, (x1, y1) (x2, y2)
Slope = Change in y (rise) y2 – y 1
Change in x (run) x2 – x1
y = mx + b m = slope b = y-intercept
Parallel lines have equal slopes
Ex. y = 5x + 4 y = 5x – 2 5 = 5
Perpendicular lines have slopes that are the negative reciprocals
Ex. y = 5x + 4 y = - 1 x – 2 5 , - 1
5 1 5
Any line has infinitely many lines parallel to it, but there is exactly one parallel line through a specified y-intercept.
Parallel Postulate: Given a line m and a point P not on line m, there is exactly one line through P and parallel to m.
Any line has infinitely many lines perpendicular to it, but there is exactly one perpendicular line through a specified y-intercept.
Perpendicular Postulate: Given a line m and a point P not on line m, there is exactly one line through P and perpendicular to m.
Similar Right Triangles
Arithmetic mean—mathematical average of a set of n numbers; to calculate it, add up all numbers and divide them by n
Ex. 16 + 18 + 48 = 82, 82 ÷ 3 = 27.333…, the average is 27.333… Geometric mean—the central tendency of a set of n numbers; to calculate it,
multiply all the numbers and take the nth root For 2 numbers, take the square (2nd) root For 3 numbers, take the cube (3rd) root For 4 numbers, take the fourth (4th) root For 5 numbers, take the fifth (5th) root
Ex. 16 x 18 x 48 = 13,824 ; the 3rd root and geometric mean is 24
A right triangle cut by an altitude will create three similar triangles
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