Postulates and Theorems Ch2 & 3 Jacob's Geometry
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- Postulate 1 (line)
- Two Points determine a line (p61)
- Postulate 2 (plane)
- Three noncollinear points determine a plane. (p61)
- Theorem: The Pythagorean Theorem
- The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides (p65)
- Theorem: The Triangle Angle Sum Theorem.
- The sum of the angles of a triangle is 180 degrees. (p.66)
- Theorem: Diameter of a circle
- If the diameter of a circle is "d", its circumference is pi(d) (p66)
- Theorem: Radius of a circle
- If the radius of a circle is "r", its area is pi times r squared.
- Postulate 3: The Ruler Postulate.
- The points on a line can be numbered so that positive number differences measure distances. (p85)
- Postulate 4: The Protractor Postulate
- The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles. (p92)
- Theorem 1: The Betweenness of Points Theorem.
- If A-B-C, THEN AB+BC=AC. (p86)
- Theorem 2: The Betweenness of Rays Theorem.
- If 0A-0B-OC, then Angle AOB + ANGLE BOC = ANGLE AOC. (p93)
- Theorem 3: Complements of the same angle
- Complements of the same angle are equal (p106)
- Theorem 4: Supplements of the same angle
- Supplements of the same angle are equal.
- Theorem 5: The angles in a linear pair
- The angles in a linear pair are supplementary. (p111)
- Theorem 7: Perpendicular lines form...
- Perpendicular lines form right angles. (p118)
- Theorem 6: Vertical angles
- Vertical angles are equal. (p112)
- Corollary to the Ruler Postulate
- A line segment has exactly one midpoint. (p99)
- Corollary to the Protractor Postulate.
- An angle has exactly one ray that bisects it. (p100)
- Corollary to the definition of a right angle.
- All right angles are equal. (p118)
- Theorem 8: If the angles in a linear pair are equal...
- If the angles in a linear pair are equal, then their sides are perpendicular.
- Definition: Supplementary
-
Two angles are supplementary iff their sum is 180 degrees.
Each angle is called the SUPPLEMENT of the other. - Definition: Complementary
- Two angles are complementary iff their sum is 90 degrees. Each angle is called the COMPLEMENT of the other.
- What is a corollary?
- A corollary is a theorem that can be easily proved as a consequence of a postulate or another theorem.
- Definition: midpoint of a line segment
- A point is the midpoint of a line segment iff it divides the line segment into two equal segments.
- Definition: bisects an angle
- A line bisects an angle iff it divides the angle into two equal angles.
- Congruent
- coinciding exactly when superimposed.
- Definition: (Betweeness of Points)
- 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.)
- The Reflexive Property
- a = a (Any number is equal to itself)
- The Substitution Property
- If a = b, then a can be substituted for b in any expression.
- The Addition Property
- If a = b, then a + c = b + c.
- The Subtraction Property
- If a = b,then a - c = b - c
- The Multiplication Property
- If a = b, then ac = bc
- The Division Property
- If a = b and c does NOT = 0, then a divided by c = b divided by c.
- The Distributive Rule
-
relates the operations of multiplication and addition. For any numbers a, b, and c,
a(b + c) = ab + ac
a(b - c) = ab - ac - Postulate
- A postulate is a statement that is assumed to be true without proof.
- Direct proof
-
If a, then b.
If b, then c.
If c, then d.
Therefore, if a, then d. - Indirect proof
-
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. - Syllogism
-
A syllogism is an argument of the form
If a then b
If b then c
Therefore, if a then c. - Conditional statements
- A conditional statement consists of two clauses, one of which begins withthe work "if" or "when" or some equivalent word.
- Hypothesis
- If a, then b. The letter a represtns the "if" clause, or hypothesis.
- Conclusion
- If a, then be. The letter b represents the "then" clause, or conclusion. (The word "then" is often omitted.)
- Converse
- 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"
- Is the converse always true?
- The converse may or may not be true, however the converse of a definition is always true.
- Theorem
- A theorem is a statement that is proved by reasoning deductively from already accepted statements.
- Premises of the argument
- The statements "if a then b, if b then c, ...."
- Conclusion of the argument
- If a then n. The conclution might be considered a theorem.