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Concepts 6-9 Notes
Reasoning and Proof Building
Concept 6 – Inductive Reasoning (Section 1.1) Inductive Reasoning: Conjecture: Counterexample: Concept 6 – Conditional Statements (Section 2.5) Conditional Statement: Hypothesis: Conclusion: Converse: Example: Use the statement below to answer each question. If you study for the test, then you will get a good grade. a) What is the hypothesis of the statement? b) What is the conclusion of the statement? c) What is the converse of the statement? Biconditional: Example: Write the converse of the statement below. If it is true, write a biconditional statement. Conditional -­‐ If two angles have the same measure, then the angles are congruent. Converse -­‐ Biconditional – Good Definition: Concept 6 – Apply Deductive Reasoning (Section 2.5) Deductive Reasoning: Law of Detachment Law of Syllogism If p → q and q → r are true statements, then p → r is also a true statement. Concept 7 – Reasoning in Algebra (Section 2.6) Algebraic Properties of Equality Property Example Addition Property If a = b , then a + c = b + c . Subtraction Property If a = b , then a − c = b − c . Multiplication Property If a = b , then ac = bc . Division Property If a = b , then
Substitution Property If a = b , then a can be substituted
for b in any expression or equation. Distributive Property a(b + c) = ab + ac a b
= c c
Properties of Equality and Congruence Property Example Explanation Reflexive Property of Equality a = a Reflexive Property of Congruence RT ≅ RT or ∠5 ≅ ∠5 Symmetric Property of Equality If a = b , then b = a . Symmetric Property of Congruence If LM ≅ RT , then RT ≅ LM . Transitive Property of Equality If a = b and b = c , then a = c . Transitive Property of Congruence If ∠A ≅ ∠B and ∠B ≅ ∠C , then
∠A ≅ ∠C . Concept 7 – Proofs Proof: 2-­‐Column Proof Each statement must have a reason to justify it. Properties, postulates, definitions, and theorems are used as reasons in a proof. Postulate: Theorem: Some Helpful Postulates and Theorems for using in proofs: Postulate 5 – Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB + BC = AB. Postulate 6 – Angle Addition Postulate If point B is in the interior of ∠ AOC,
then m∠ AOB + m∠ BOC = m∠ AOC.
Theorem 3.1 - Right Angles Congruence Theorem Theorem 2.1 -­‐ Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then________________________________________.
Theorem 2.2 -­‐ Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then________________________________________ . Postulate 7 – Linear Pair Postulate Theorem 2.3 -­‐ Vertical Angles Theorem Concept 8 – Relationships Between Lines (Section 3.1) Pairs of Lines/Segments Parallel Lines: Perpendicular Lines: Skew Lines: Parallel Planes: Concept 8 – Theorems about Perpendicular Lines (Section 3.2) Theorem 3.2 If two lines are perpendicular, then they intersect to form __________________________________________. Theorem 3.3 If two lines intersect to form a linear pair of congruent angles, then the lines are _______________________. Theorem 3.4 If two sides of two adjacent acute angles are perpendicular, then the angles are _________________________. Concept 8 – Angles Formed by Transversals (Section 3.3) Transversal: Corresponding Angles Alternate Interior Angles Same-­‐Side Interior Angles (Consecutive Interior) Alternate Exterior Angles Concept 8 – Parallel Lines and Transversals (Section 3.4) Angle Pairs and Parallel Lines *Use these Theorems as reasons for why two angles are congruent or supplementary* Postulate 8 -­‐ Corresponding Angles Postulate If a transversal intersects two parallel lines, then corresponding angles are _________________. Theorem 3.5 -­‐ Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are _________________. Theorem 3.6 -­‐ Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are _________________. Theorem 3.7 – Same-­‐Side Interior Angles Theorem If a transversal intersects two parallel lines, then same-­‐side interior angles are _________________. Concept 8 – Showing Lines are Parallel (Section 3.5 and 3.6) Theorems to Prove Lines are Parallel *Use these Theorems as reasons for how you know two lines are parallel* Postulate 9 -­‐ Corresponding Angles Converse If 2 lines are cut by a transversal so that corresponding angles are congruent, then lines are parallel. Theorem 3.8 -­‐ Alternate Interior Angles Converse If 2 lines are cut by a transversal so that Alternate Interior Angles are congruent, then lines are parallel Theorem 3.9 -­‐ Alternate Exterior Angles Converse If 2 lines are cut by a transversal so that Alternate Exterior Angles are congruent, then lines are parallel. Theorem 3.10 – Same-­‐Side Interior Angles Converse If 2 lines are cut by a transversal so that Same-­‐Side Interior Angles are supplementary, then lines are parallel. Theorem 3.11 – Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are ________________________________________. Theorem 3.12 – Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line,
then they are ____________________________________. Concept 9 – Slopes of Parallel and Perpendicular Lines Slope = rise y2 − y1
=
run x2 − x1
Slope in the Coordinate Plane
- Negative slope lines ________________________
-­‐ Positive slope lines ________________________________ -­‐ Zero slope lines ____________________________________ -­‐ Undefined slope lines _____________________________ Parallel Lines
If two non-vertical lines are parallel, then __________________________________________________.
Any two vertical lines are parallel.
Perpendicular Lines
If two non-vertical lines are perpendicular, then ________________________________________________.
Any horizontal line and vertical line are perpendicular.
m=
b=
To find the x-intercept:
- substitute zero in for y
m=
(x1, y1) =
To find the y-intercept:
- substitute zero in for x
Example: Write the equation of a line in slope-­‐intercept for that meets the given conditions. 1
2
1. Through (2,3) and parallel to y = x + 8
2. Through (-4,5) and perpendicular to y = − x −12 2
3