Postulates and Theorems(Prashanth Gowda)
The reason I chose the following theorems is because they all have to do with Unit 1, and they all help you understand and solve geometrical problems about the length of lines and measure of angles. All these theorems help us understand lines, point, angles, and planes.
If two points lie in a plane, then the line containing those points lies in the plane. Line AB lies in Plane ABC. If two unique lines intersect, then they intersect at exactly one point. Line AB and Line GH intersect at Point D. If two unique planes intersect, then they intersect at exactly one line. Plane Q and Plane P intersect at Line AB. Lesson 2 Law of Detachment: If p → q is a true statement and p is true, then q is true. Given: If M is the midpoint of a segment, then Point M bisects the segment. M is the midpoint of Segment AB. Conjecture: Segment AM is congruent to segment MB If M is the midpoint of a segment, then Point M bisects the segment. p → q The given statement M is the midpoint of Segment AB matches the hypothesis. P and Q are both true. By the Law of Detachment, Segment AM is congruent to Segment MB. Law of Syllogism: If p → q and q → r are true statements, then p → r is a true statement.
Given: If 90 degrees < m<A which <180 degrees, then Angle A is obtuse. If Angle A is obtuse, then it is not acute. Conjecture: If 90 degrees < m<A which <180 degrees, then Angle A is not acute. p: The measure of an angle is between 90 degrees and 180 degrees. q: The angle is obtuse. r: The angle is not acute. It's given that p → q and q → r. Thus p → r since p → q and q → r are true statements. The conjecture is valid by the Law of Syllogism. |
Lesson 3
Law of Contrapositive: A conditional and its contrapositive are logically equivalent.
Example: If an animal is living, then it's breathing.(Conditional)
If the animal is not breathing, then it's not living.(Contrapositive)
Law of Contrapositive: A conditional and its contrapositive are logically equivalent.
Example: If an animal is living, then it's breathing.(Conditional)
If the animal is not breathing, then it's not living.(Contrapositive)
Lesson 4
Distributive Property- A(B+C) = AB+AC
Properties of Equality
Addition Property of Equality- If a=b, then a+c=b+c
Subtraction Property of Equality- If a=b, then a-c=b-c
Multiplication Property of Equality - If a=b, then ac=bc
Division Property of Equality- If a=b then a/c=b/c
Symmetric Property of Equality-A+B=B+A
Transitive Property of Equality- If a=b and b=c, then a=c.
Reflexive Property of Equality- A=A
Substitution Property of Equality- If a=b and a+10=20, then b+10=20
Properties of Congruence
Symmetric Property of Congruence-A+B=B+A
Transitive Property of Congruence- If a=b and b=c, then a=c.
Reflexive Property of Congruence- A=A
Lesson 6
Segment Addition Postulate- If Point M is between Point A and Point B, then AM + MB = AB.
If AM + MB = AB, then Point M is somewhere between Point A and B.
Midpoint Theorem- If Point M is the midpoint of Line Segment AB, then AM = (AB)/2 and MB = (AB)/2.
Ruler Postulate- The distance between A and B, written as AB, is the absolute value of the difference between the coordinates of A and B.
Example: If the coordinates of Point A is (2,0) and the coordinates of B is (6,0), then AB = 4.
Distributive Property- A(B+C) = AB+AC
Properties of Equality
Addition Property of Equality- If a=b, then a+c=b+c
Subtraction Property of Equality- If a=b, then a-c=b-c
Multiplication Property of Equality - If a=b, then ac=bc
Division Property of Equality- If a=b then a/c=b/c
Symmetric Property of Equality-A+B=B+A
Transitive Property of Equality- If a=b and b=c, then a=c.
Reflexive Property of Equality- A=A
Substitution Property of Equality- If a=b and a+10=20, then b+10=20
Properties of Congruence
Symmetric Property of Congruence-A+B=B+A
Transitive Property of Congruence- If a=b and b=c, then a=c.
Reflexive Property of Congruence- A=A
Lesson 6
Segment Addition Postulate- If Point M is between Point A and Point B, then AM + MB = AB.
If AM + MB = AB, then Point M is somewhere between Point A and B.
Midpoint Theorem- If Point M is the midpoint of Line Segment AB, then AM = (AB)/2 and MB = (AB)/2.
Ruler Postulate- The distance between A and B, written as AB, is the absolute value of the difference between the coordinates of A and B.
Example: If the coordinates of Point A is (2,0) and the coordinates of B is (6,0), then AB = 4.
Lesson 7
Vertical Angle Theorem- If two angles are vertical angles, then they are congruent. Angle 4 is congruent to Angle 2. Angle Addition Postulate- If Point B is in the interior of angle AOC, then m<AOB+m<BOC=m<AOC. Protractor Postulate- Given Ray AB and a point O on Ray AB , all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180. Lesson 8
Linear Pair Theorem- If two angles form a linear pair, then they are supplementary. Angle A and Y are supplementary. Congruent Supplements Theorem- If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. Angle 1 and Angle 3 are both supplementary to Angle 2. Congruent Supplements Theorem(Part2)-If two congruent angles are supplementary, then each angle is a right angle. Angle ABD and ADC are both right angles. Right Angle Congruence Theorem- All right angles are congruent. Both right angles equal 90 degrees, therefore they are congruent. Congruent Complements Theorem- If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. Angle 4 and 6 are both complementary to Angle 5. Common Segments Theorem- Given collinear points A, B, C, and D arranged on a segment such that A and D are the endpoints, B is between A and C, and C is between B and D. If Segment AC is congruent to Segment BD, then Segment AB is congruent to segment CD. |