Geometry/Trigonometry Unit 2: Parallel Lines Notes Period:
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1 Geometry/Trigonometry Unit 2: Parallel Lines Notes Name: Date: Period: # (1) Pg #1-10 all (2) Pg #12-22 Even and 30, 32 (3) Pg 114 #1-6; 9-13 (4) Pg #15-18; 20; 22; 24; 26; 29 and 30***Quiz Tomorrow*** (5) Pg 123 #1-5 (6) Pg #6-24 Even (7) Pg 129 #1 8 (8) Pg #9-12, 16 18; FF #20 (9) Pg 133 #1 12 all *****Quiz Tomorrow***** (10) Pg 137 #1 10 (11) Pg #12 22 Even (12) Pg 144 #1 8 (13) Pg #9-17; 19 *****Quiz Tomorrow***** (14) Pg #2 18 even, 23-26; (15) Pg 157 #1 20, skip #14 18 *****Test Tomorrow*****
2 Geometry Notes 3.1 Exploring Lines and Planes Parallel Lines coplanar lines that do Intersecting lines coplanar lines that have in common Perpendicular if they meet lines that intersect but not at a right angle Theorem 3.1 Transitivity of Parallel lines: If two lines are parallel to the same line, then they are parallel to each other. (coplanar or non-coplanar lines) Theorem 3.2 Property of Perpendicular Lines: If two coplanar lines are perpendicular to the same line, then they are parallel to each other. (coplanar only) Skew lines that do not lie in the Parallel planes planes that intersect. A good example is the floor and the ceiling in this room. To draw parallel planes, use two parallelograms: To draw intersecting planes, follow these steps:
3 Geometry Notes 3.2 Connections to Algebra Systems of Equations Exactly one solution - (Intersecting Lines) No Solution - (Parallel Lines) IMS Infinitely Many Solutions - (Same Line coincident) Postulate 12 If two distinct lines intersect, then their intersection is exactly one point Parallel and Perpendicular Postulates Postulate 13 Parallel Postulate: If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Postulate 14 Perpendicular Postulate: If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Solving Systems Refresher: (Ex.1) Substitution Method:
4 (Ex.2) Elimination Method: Writing equations of lines refresher: (Ex3.) Write an equation of the line that is parallel to the line point (6, -12). and passes through the (Ex4.) Write an equation of the line that is perpendicular to the line the point (7, -3). and passes through
5 Geometry Notes 3.3 Using the Laws of Logic Using Consider the conditional statement: If the weather is good, then I will go swimming. The negation of a hypothesis or a conclusion is formed by denying the original hypothesis or conclusion. Statement Symbol Negation Symbol The weather is good p I will go swimming q The of the conditional statement is. That is if not q, then not p. The contrapositive of a statement is IFF the conditional statement is (Ex.1) If it is Halloween, then I will get candy from my teacher. If I don t get candy from my teacher, then it is not Halloween. If I get candy from my teacher, then it is Halloween. : In mathematical logic, the Law of Syllogism says that if the following two statements are true: (1) If p, then q. (2) If q, then r. Then we can derive a third true statement: (3) If p, then r. (Ex2.) If the following statements are true, use the Law of Syllogism to derive a new true statement. 1) If it snows today, then I will wear my gloves. 2) If I wear my gloves, my fingers will get itchy. Let p be the statement "it snows today", let q be the statement "I wear my gloves", and let r be the statement "my fingers get itchy". Then (1) and (2) can be written 1) If p, then q. 2) If q, then r. So, by the Law of Syllogism, we can deduce 3) If p, then r or
6 In mathematical logic, the Law of Detachment says that if the following two statements are true: (1) If p, then q. (2) p Then we can derive a third true statement: (3) q. (Ex.3) If the following statements are true, use the Law of Detachment to derive a new true statement. 1) If you are a penguin, then you live in the Southern Hemisphere. 2) You are a penguin. Let p be the statement "you are a penguin", let q be the statement "you live in the Southern Hemisphere". Then (1) and (2) can be written 1) p 2) If p, then q. So, by the Law of Detachment, we can deduce that q is true. That is, Geometry Notes 3.4 Styles of Proofs Coordinate Proof a way to prove something by a over it. Paragraph Proof a. It explains the reasoning that a person might use to convince the reader that a statement is true. Theorem If two lines are perpendicular, then they intersect to form four right angles. Theorem All right angles are congruent. Theorem 3.5 If two lines intersect to form a pair of adjacent congruent angles, then the lines are perpendicular. Flow Proof uses to show the flow or. A is written each statement. Two-Column Proof (aka Formal Proof) a of statements, each with a. implies a logical order. ***The type we will use most often***
7 Geometry Notes 3.5 Properties of Parallel Lines A is a line that intersects two or more coplanar lines at different points. Angles formed by a transversal: In the figure, the transversal t intersects the lines l and m Two angles are if they occupy corresponding positions such as Two angles are t if they lie between l and m on opposite sides of t, such as Two angles are if they lie outside of l and m on opposite sides of t, such as m l Two angles are, if they lie between l and m on the same side of t, such as t Example: Naming Pairs of Angles Corresponding Angles: l Alternate Interior Angles: Alternate Exterior Angles: m Consecutive Interior Angles (Same Side Interior Angles):
8 Postulate 15 - Corresponding Angles Postulate - If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. ****IMPORTANT NOTE**** the hypothesis states the lines MUST BE PARALLEL!!! Theorem Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Theorem Consecutive Interior Angles Theorem: If two pairs of parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Theorem Alternate Exterior Angles Theorem: If two pairs of parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the second. Geometry Notes 3.6 Proving Lines are Parallel The following postulates and theorems and the converses of those discussed in section 3.5 Postulate 16 - Corresponding Angles Converse: If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Theorem Alternate Interior Angles Converse: If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Theorem Consecutive Interior Angles Converse: If two lines are cut by a transversal so that consecutive interior angles are supplementary, the lines are parallel. Theorem Alternate Exterior Angles Converse: If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.
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