Geometry Property Postulates and defintions
A Quantity Is Congruent (Equal) to itself.
Anything is equal to itself. In symbols,
A = A. or
If two things are equal to a third thing, then they
are equal to one another.
If A = B and B = C, then A = C.
If one value (a) is equal to another value (b), which is equal to another value (c), then the first value (a) is equal to the third value
If <a = <b and <b = <c then <a = <c.
If equal quantities are added to equal quantities then the sums are equal
If A = B, then A + C = B + C.
If equal quantities are subtracted from equal quantities then the differences are equal.
If A = B, then A - C = B - C.
If equal quantities are multiplied by equal quantities then the products are equal
If A = B, then A x C = B x C.
If equal quantities are divided by equal nonzero quantities then the quotions are equal
If A = B, then A / C = B / C.
A quantity and its equal(s) are interchangeable in an expression. If abc=def is given, and ab+de=abc, then ab+de=def would be true.
The whole is equal to the sum of its parts.
Also: Betweeness of points AB + BC AC
Angle Addition Postulate: m<ABC + m<CBD = m<ABD
Two Points determine a straight line
Construction Straight line
From a given point one and only one perpendicular can be drawn to the line.
All Right angles are congruent. All right angles = 90 degrees. Formed by 2 lines perprndicular to each other
An angle whose sides lie in opposite directions from the vertex in the same straight line and which equals two right angles
If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent
m<A + m<B = 180 Degrees and m<A + m<C = 180 Degrees Then m<B + m<C = 180 Degrees and <B = <C
If two angles are compliments of the same angle (or of congruent angles), then the two angles are congruent. Compliments of the same angle or congruent angles are congruent
IF m<A + m<B = 90 Drgrees and m<B + m<C = 90 Degrees Then m<B + m<C = 90 Degrees<B = <C
If two angles form a linear pair,then they are supplementary (m < 1 + m< 2 = 180 Degrees). Two angles that are adjacent (share a leg) and supplementary (add up to 180°)
Vertical angles are on opposite sides of the X formed when two lines intersect Vertical Angles are congruent.They are congruent Vertical angles are always congruent, or of equal measure. Both pairs of vertical angles (four angles altogether) always sum to a full angle (360°).
The sum of the interior angles of a triagle is 180 Degrees
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle
If two sides of a triangle are congruent, the angles opposite are congruent
Base Angle Theorem
If two parallel lines are cut by a transversal, and the corresponding angles are congruent the lines are parallel
Corresponding Angles Converse
If two lines are cut by a transversal, then the pairs of corresponding angles are congruent
If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
<5 = <6
Alternate Interior Angles
Formed when two lines are cut by another line (transversal), the pairs of angles formed outside the two lines and on the opposite of the transversal.
If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
AB || CD and EF is the transversal cutting the parallel lines. So, ∠2 and ∠8;and ∠1 and ∠7 are the pairs of alternate exterior angles in the figure shown.
Alternate Exterior Angles
If two parallel lines are cut by a transversal,the interior angles on the same side of the transversal are supplemetary.
<4 and < 5 are Interior Same side angles so are <3 and <6
Interiors on Same Side Angles
If two lines are cut by a transversal and the alternate interior angles are congruent then the lines are parallel
Transversal EFGH intersects lines AB and CD such that a pair of alternate E are equal.
Since the corresponding angles of the lines AB and CD are equal, the lines are parallel.
Alternate Interior Angles Converse
If two lines are cut by a transversal and the alternate exterior angles are congruent then the lines are parallel or
If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel.
Alternate Exterior Angles Converse
If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary then the lines are parallelor If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel.
Interiors on Same Side Converse
Any angle whose measure is less than 90 degrees. Does not include 0 or 90 Degrees
Angles that share a vertex and that have a side in common.
A point that divides a segment into two congruent segments. The point on that line segment that divides the segment two congruent segments.
Forming right angles. Right angles measure 90 degrees
A geometic statement assumed to be true without proof
Angles on the same side of the transversal, and they’re outside the parallel lines. <7 and <1 also <8 and <2
Two same-side exterior angles are supplementary.
m<7 + m<1 = 180 degrees and m<2 + m<8 = 180 Degrees
Same-Sided Exterior Angles
If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.
If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.
Same-Sided Interior Angles
A line that crosses two parallel lines
Congruent anglea formed by two intersectin lines that make an X; the vertical angles are on opposite sides of the X.
In the diagram we have AX = 5 and XB = 6. AB = 5 + 6 = 11.
Segment Addition Postulate
States that when both sides of an equation have the same number subtracted from them, the remaining expressions are still equal.
If 12x = 24x/2, then 12x - 7 = 24x/2 - 7.
If a = b, then a - c = b - c.
Subtraction Property of Equality
If we have an equation like
3x - 5 = y
we can rewrite that as
y = 3x - 5
and it is still true.
If a number is equal to a second number, then the second number is also equal to the first.
if 13 – 5 = 8 then 8 = 13 – 5.
If 4•5 = 20 then 20 = 4•5.
Symmetric Property of Equality
If I know that two variables both refer
to the same number (that's what "equal" means), then anything I say about one of them is also true of the other
If a = b, then b can replace a in any equation without changing the truth value of the equation. For example:
Let a = b.
Let d = a + 2.
Then d = b + 2.
Substitution Poperty of Eqrality
This form of a conditional statement is obtained by interchanging the hypothesis and conclusion. If it is the first of the month then the rent is due becomes If the rent is due then it is the first of the month
Converse Conditional Statement
This form of a conditional statement is obtained by negating the hypothesis and negating the conclusion. If it is the first of the month then the rent is due becomes If it is not the first of the month then the rent is not due
Inverse Conditional Statement
This form of a conditional statement is obtained by interchanging and negating the hypothesis and conclusion. If it is the first of the month then the rent is due becomes If the rent is not due then it is not the first of the month
Contrapositive Conditional Statement
An instance of a form of reasoning in which a conclusion is drawn (whether validly or not) from two given or assumed propositions (premises), each of which shares a term with the conclusion, and shares a common or middle term not present in the conclusion (e.g., all dogs are animals; all animals have four legs; therefore all dogs have four legs ).
If q, then r. If p then q is true and if q then r is true then p then r is also true.
a diagram that shows all possible logical relations between a finite collection of sets.The diagram shows and or or nor relationships
The "If" part of a statement. A statement that might be true, which can then be tested
The "then" part of a conditional statement.
Name for an If...Then statementA compound statement formed by joining two statements with "if" and "then"
A declarative statement which is either true or false, but never both
Sometimes in mathematics it's important to determine what the opposite of a given mathematical statement. If a statement is true, then its opposite is false (and if a statement is false, then its opposite is true).
usually be formed by placing the word "not" into the original statement. The negation will always have the opposite truth value of the original statement.
Under negation, what was TRUE, will become FALSE -
or - what was FALSE, will become TRUE.
A compound statement resulting from the joining of two statements using an AND. Both statements must be true for the conjunction to be true.
n logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false
Conjunction A and B
A compound statement resulting from the joining of two statements using OR.If both statements are false Then the compound is false.Logical OR is also known as logical disjunction. Disjunction literally means "a state of being disjoined". Logical disjunction holds when at least one of the given conditions is true. It is a binary operator which requires at least two operands for its application.
A statement of a mathematical fact that can be proved
A form of deductive reasoning that is used to draw conclusions from true conditional statements. If p then q is true then p is true and q is true.
Syllogism works with the statments "P implies Q" and "Q implies R" (notice Q is mentioned twice). If both those statements are true, then the statement "P implies R" is true.
The law of Detachment
if a number is equal to a second number, and the second number is equal to a third, then the first number and the third number are also equal.
If 3 + 7 = 10 and 10 = 4 + 6, then 3 + 7 = 4 + 6.
If 22 / 11 = 2 and 2 = 18 / 9, then 22 / 11 = 18 / 9.
Transitive property of equality:
Defined as two exterior angles on opposite sides of a transversal which lie on different parallel lines.
ngles 1 and 8 (and angles 2 and 7) are called alternate exterior angles. They’re on opposite sides of the transversal, and they’re outside the parallel lines.
Alternate Exterior Angles
two interior angles which lie on different parallel lines and on opposite sides of a transversal.he pair of angles 3 and 6 (as well as 4 and 5) are alternate interior angles. These angle pairs are on opposite (alternate) sides of the transversal and are in between (in the interior of) the parallel lines.
Alternate Interior Angles
In the figure above, angles 4 and 6 are consecutive interior angles. So are angles 3 and 5. Consecutive interior angles are supplementary. The pairs of angles on one side of the transversal but inside the two lines are called Consecutive Interior Angles.
Consecutive Interior Angles
Exactly equal in size and shape. Congruent sides or segments have the exact same length. Congruent angles have the exact same measure. For any set of congruent geometric figures, corresponding sides, angles, faces, etc. are congruent.
Lines or curves that all intersect at a single point.
Identical, one superimposed on the other. That is, two or more geometric figures that share all points. For example, two coincident lines would look like one line since one is on top of the other.
Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining."
Note: As in the example, the contrapositive of any true proposition is also true.
Switching the hypothesis and conclusion of a conditional statement. For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining."
Note: As in the example, a proposition may be true but have a false converse.
Negating both the hypothesis and conclusion of a conditional statement. For example, the inverse of "If it is raining then the grass is wet" is "If it is not raining then the grass is not wet".
Note: As in the example, a proposition may be true but its inverse may be false.
Inverse of a Conditional
A way of writing two conditionals at once: both a conditional and its converse.
For example, the statement "A triangle is equilateral iff its angles all measure 60°" means both "If a triangle is equilateral then its angles all measure 60°" and "If all the angles of a triangle measure 60° then the triangle is equilateral".
if and only if
Two features that are situated the same way in different objects.
A pair of adjacent angles formed by intersecting lines. Angles 1 and 2 below are a linear pair. So are angles 2 and 4, angles 3 and 4, and angles 1 and 3. Linear pairs of angles are supplementary.
Linear Pair of Angles
Two distinct coplanar lines that do not intersect. Note: Parallel lines have the same slope.
A 180° angle.
A line that cuts across a set of lines or the sides of a plane figure. Transversals often cut across parallel lines.
An angle on the interior of a plane figure.
Examples: The angles labeled 1, 2, 3, 4, and 5 in the pentagon are all interior angles. Angles 3, 4, 5, and 6 in the second example below are all interior angles as well (parallel lines cut by a transversal).
Note: The sum of the interior angles of an n-gon is given by the formula (n – 2)·180°. For a triangle this sum is 180°, a quadrilateral 360°, a pentagon 540°, etc.
Angle congruence is reflexive, symmetric, and transitive. Here are some examples.
REFLEXIVE: For any angle <A = <A
TRANSITIVE: If <A = <B and < B = <C then <A = <C
SYMMETRIC: If <A = <B then <B = <A
Properties of Angle Congruence