Postulates And Theorems Ch2 & 3 Jacob's Geometry

Front 1

Postulate 1 (line)

Back 1

Two Points determine a line (p61)

Front 2

Postulate 2 (plane)

Back 2

Three noncollinear points determine a plane. (p61)

Front 3

Theorem: The Pythagorean Theorem

Back 3

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides (p65)

Front 4

Theorem: The Triangle Angle Sum Theorem.

Back 4

The sum of the angles of a triangle is 180 degrees. (p.66)

Front 5

Theorem: Diameter of a circle

Back 5

If the diameter of a circle is "d", its circumference is pi(d) (p66)

Front 6

Theorem: Radius of a circle

Back 6

If the radius of a circle is "r", its area is pi times r squared.

Front 7

Postulate 3: The Ruler Postulate.

Back 7

The points on a line can be numbered so that positive number differences measure distances. (p85)

Front 8

Postulate 4: The Protractor Postulate

Back 8

The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles. (p92)

Front 9

Theorem 1: The Betweenness of Points Theorem.

Back 9

If A-B-C, THEN AB+BC=AC. (p86)

Front 10

Theorem 2: The Betweenness of Rays Theorem.

Back 10

If 0A-0B-OC, then Angle AOB + ANGLE BOC = ANGLE AOC. (p93)

Front 11

Theorem 3: Complements of the same angle

Back 11

Complements of the same angle are equal (p106)

Front 12

Theorem 4: Supplements of the same angle

Back 12

Supplements of the same angle are equal.

Front 13

Theorem 5: The angles in a linear pair

Back 13

The angles in a linear pair are supplementary. (p111)

Front 14

Theorem 7: Perpendicular lines form...

Back 14

Perpendicular lines form right angles. (p118)

Front 15

Theorem 6: Vertical angles

Back 15

Vertical angles are equal. (p112)

Front 16

Corollary to the Ruler Postulate

Back 16

A line segment has exactly one midpoint. (p99)

Front 17

Corollary to the Protractor Postulate.

Back 17

An angle has exactly one ray that bisects it. (p100)

Front 18

Corollary to the definition of a right angle.

Back 18

All right angles are equal. (p118)

Front 19

Theorem 8: If the angles in a linear pair are equal...

Back 19

If the angles in a linear pair are equal, then their sides are perpendicular.

Front 20

Definition: Supplementary

Back 20

Two angles are supplementary iff their sum is 180 degrees.
Each angle is called the SUPPLEMENT of the other.

Front 21

Definition: Complementary

Back 21

Two angles are complementary iff their sum is 90 degrees. Each angle is called the COMPLEMENT of the other.

Front 22

What is a corollary?

Back 22

A corollary is a theorem that can be easily proved as a consequence of a postulate or another theorem.

Front 23

Definition: midpoint of a line segment

Back 23

A point is the midpoint of a line segment iff it divides the line segment into two equal segments.

Front 24

Definition: bisects an angle

Back 24

A line bisects an angle iff it divides the angle into two equal angles.

Front 25

Congruent

Back 25

coinciding exactly when superimposed.

Front 26

Definition: (Betweeness of Points)

Back 26

A point is between two other points on the same line iff its coordinate is between their coordinates. ( A-B-C iff a<b<c or a>b>c.)

Front 27

The Reflexive Property

Back 27

a = a (Any number is equal to itself)

Front 28

The Substitution Property

Back 28

If a = b, then a can be substituted for b in any expression.

Front 29

The Addition Property

Back 29

If a = b, then a + c = b + c.

Front 30

The Subtraction Property

Back 30

If a = b,then a - c = b - c

Front 31

The Multiplication Property

Back 31

If a = b, then ac = bc

Front 32

The Division Property

Back 32

If a = b and c does NOT = 0, then a divided by c = b divided by c.

Front 33

The Distributive Rule

Back 33

relates the operations of multiplication and addition. For any numbers a, b, and c,
a(b + c) = ab + ac
a(b - c) = ab - ac

Front 34

Postulate

Back 34

A postulate is a statement that is assumed to be true without proof.

Front 35

Direct proof

Back 35

If a, then b.
If b, then c.
If c, then d.
Therefore, if a, then d.

Front 36

Indirect proof

Back 36

Suppose not d is true.
If not d, then e
If e, then f,
and so on, until we come to a contradiction.
Therefore, not d is false; so d is true.

Front 37

Syllogism

Back 37

A syllogism is an argument of the form
If a then b
If b then c
Therefore, if a then c.

Front 38

Conditional statements

Back 38

A conditional statement consists of two clauses, one of which begins withthe work "if" or "when" or some equivalent word.

Front 39

Hypothesis

Back 39

If a, then b. The letter a represtns the "if" clause, or hypothesis.

Front 40

Conclusion

Back 40

If a, then be. The letter b represents the "then" clause, or conclusion. (The word "then" is often omitted.)

Front 41

Converse

Back 41

The converse of a conditional statement is found by interchanging the hypothesis and conclusion. The converse of "if a then b" is "if b then a"

Front 42

Is the converse always true?

Back 42

The converse may or may not be true, however the converse of a definition is always true.

Front 43

Theorem

Back 43

A theorem is a statement that is proved by reasoning deductively from already accepted statements.

Front 44

Premises of the argument

Back 44

The statements "if a then b, if b then c, ...."

Front 45

Conclusion of the argument

Back 45

If a then n. The conclution might be considered a theorem.