Chapter 3 – Parallel and Perpendicular Lines In this chapter we address three Big IDEAS: Using properties of parallel and perpendicular lines Proving relationships using angle measures Making connections to lines in algebra Section: 3 – 1 Identify Pairs of Lines and Angles Essential Question What angles pairs are formed by two lines and a transversal? Warm Up: Key Vocab: Parallel Lines Two lines that do not intersect and are coplanar Skew Lines Lines that do not intersect and are NOT coplanar Parallel Planes Two planes that do not intersect Transversal A line that intersects two or more coplanar lines at different points line t is a transversal Corresponding Angles Two angles that are formed by two lines and a transversal and occupy corresponding positions are corresponding angles Alternate Interior Angles Two angles that are formed by two lines and a transversal and lie between the two lines and on opposite sides of the are interior angles
Key Vocab: Paragraph Proof A type of proof written in paragraph form Postulates and Theorems: Corresponding Angles Postulate Converse IF two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. Alternate Interior Angles Theorem Converse If two lines are cut by a transversal so the alternate interior angles are congruent then the lines are parallel. Alternate Exterior Angles Theorem Converse If two lines are cut by a transversal so the then the lines are parallel.
alternate exterior angles are congruent Consecutive Interior Angles Theorem Converse If two lines are cut by a transversal so the consecutive interior angles are supplementary then the lines are parallel. Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other. If a||b and b||c, then a||c. Show: Ex 1: Find the value of y that makes a || b. 23 by the alternate exterior angles converse Ex 2: Can you prove that lines a and b are parallel? Justify your answer. a. Yes, Alt. Ext. Conv b. Yes, Corr. Conv. c. No Ex 3: Prove that if are supplementary then a || b. Given: are supplementary
Prove: a || b Steps Reasons are supplementary Given Def of Supplementary Ang’s Linear Pair Post. Add. Prop. Of Eq Commutative Prop. Of Eq Substitution Subtr Prop. Of Eq a || b Consec Int Ang’s Converse Ex 4: Prove the Alternate Interior Angles Theorem Converse in two-column form. Given: Prove: a || b Steps Reasons Given Vert. Theorem Transitive Property Corr. Postulate Converse Ex 5: Prove the Alternate Interior Angles Theorem Converse in paragraph form. Given: Prove: a || b
Horizontal lines have zero slope. Section: 3– 5 Write and Graph Equations of Lines Essential Question How do you write an equation of a line? Warm Up: Key Vocab: Slope-intercept Form A linear equation written in the form y = mx + b where m is the slope and b is the y-intercept. Best form for graphing. Standard form A linear equation written in the form Ax + By = C, where A, B and C are integers and A and B are not both zero. Point-slope Form A linear equation written in the form , where m is the slope and is a coordinate. Vertical Line where a is the x-intercept Vertical lines have undefined or no slope Horizontal Line where b is the y-intercept Horizontal lines have zero slope Show: Ex 1: Write an equation of the line in slope-intercept form. y = 4x - 3 Ex 2: Graph 2x – 3y = -12.
perpendicular, four right angles. If two sides of two adjacent acute angles are perpendicular, Then the angles are complementary. If are complementary. Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, Then it is perpendicular to the other. and Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are perpendicular to the same line, Then they are parallel to each other. Show: Ex 1: In the figure, are congruent. What can you conclude about ? Ex 2: Determine which other lines, if any, must be perpendicular. Explain your reasoning. By the Corr. Angles Post Conv.,. So, Perp. Transversal Theorem Ex 3: Is ? Is ? Explain your reasoning. Yes, by the Lines Perpendicular to a Transversal Theorem Yes,: by the Lines Perpendicular to a Transversal Theorem, so by the Perpendicular Transversal Theorem