Glencoe Geometry – Chapter 2 – Tools of Geometry

BIG Ideas:
Make conjectures, determine whether a statement is true of false, and find counterexamples for statements.
Use deductive reasoning to reach valid conclusions.
Verify algebraic and geometric conjectures using informal and formal proof. (H)
Write proofs involving segment and angle theorems. (H)

2-1 Inductive Reasoning and Conjecture

Main Ideas:
Make conjectures based on inductive reasoning.
Find counterexamples.
What you should know:
Conjecture
Inductive Reasoning
Counterexample

2-2 Logic

Main Ideas:
Determine truth values of conjunctions and disjunctions.
Construct truth tables.
What you should know:
Negation
Conjunction
Disjunction
Truth Table


2-3 Conditional Statements

Main Ideas:
Analyze statements in if-then form.
Write the converse, inverse, and contrapositive of if-then statements.
What you should know:
Conditional Statement
If-Then (Hypothesis, Conclusion)
Converse
Inverse
Contrapositive
Biconditional


2-4 Deductive Reasoning

Main Ideas:
Use the Law of Detachment
Use the Law of Syllogism
What you should know:
Deductive Reasoning
Law of Detachment
Law of Syllogism


2-5 Postulates and Paragraph Proofs
Main Ideas:
Identify and use basic postulates about points, lines, and planes.
Write paragraph proofs. (H)
What you should know:
Postulate
Segment Addition Postulate: If B is between A and C, then AB + BC = AC
Postulate 2.1: Through any two points, there is exactly one line
Postulate 2.2: Through any three points, there is exactly one plane
Postulate 2.3: A line contains at least two points
Postulate 2.4: A plane contains at least three points not on the same line
Postulate 2.5: If two points line in a plane, then the entire line containing those points lies in that plane
Postulate 2.6: If two lines intersect, then their intersection is exactly one point.
Postulate 2.7: If two planes intersect, then their intersection is a line.
Theorem
Theorem 2.1: If M is the midpoint of external image 0clip_image002.png, then external image 0clip_image004.png.

2-6 Algebraic Proofs

Main Ideas:
Use Algebra to write two column proofs. (H)
Use properties of equality in geometry proofs. (H)
What you should know:
Reflexive Property: a = a
Symmetric Property: If a = b, then b = a
Transitive Property: If a = b and b = c, then a = c
(works with segment and angle measures)
Substitution Property
Addition and Subtraction Properties
Multiplication and Division Properties
Distributive Property

2-7 Proving Segment Relationships

Main Ideas:
Write proofs involving segment addition. (H)
Write proofs involving segment congruence. (H)
What you should know:
Postulate 2.9: If A, B, and C are collinear and B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.
Reflexive, Symmetric and Transitive Property for Segment Congruence

WEBSITE: Practice drawing conclusions from given information.


2-8 Proving Angle Relationships

Main Ideas:
Write proofs involving supplementary and complementary angles. (H)
Write proofs involving congruent and right angles. (H)
What you should know:
Postulate 2.11: ANGLE ADDITION POSTULATE: If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS. If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS.
Theorem 2.3: SUPPLEMENT THEOREM: If two angles form a linear pair, then they are supplementary.
Theorem 2.4: COMPLEMENT THEOREM: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
Reflexive, Symmetric and Transitive Property for Angle Congruence
Theorem 2.6: Angles supplementary to the same angle or to congruent angles are congruent.
Theorem 2.7: Angles complementary to the same angle or to congruent angles are congruent.
Theorem 2.8: VERTICAL ANGLES THEOREM: If two angles are vertical angles, then they are congruent.
Theorem 2.9: Perpendicular lines intersect to form four right angles.
Theorem 2.10: All right angles are congruent.
Theorem 2.11: Perpendicular lines form congruent adjacent angles.
Theorem 2.12: If two congruent angles form a linear pair, then they are right angles.