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Chapter 3 Goals – Perpendicular and Parallel Lines 3.1 Lines and Angles Identify relationships between lines such as parallel, perpendicular, or skew Identify the four types of angles formed by 2 lines and a transversal 3.2 Proof and Perpendicular Lines Compare the three different kinds of proof (2-column, paragraph, or flow) Prove results about perpendicular lines 3.3 Parallel Lines and Transversals Given that lines are parallel, prove congruent angles using theorems Using parallel line theorems, set up equations to solve for missing variables and angles (don’t forget vertical angles and linear pair!) 3.4 Proving Lines are Parallel Prove that two lines are parallel using parallel converse theorems 3.5 Using Properties of Parallel Lines Use parallel and perpendicular line theorems to determine if two lines are parallel or not. 3.6 Parallel Lines in the Coordinate Plane Rewrite equations in slope-intercept form and identify the slope of the line. Using slopes, determine if two lines are parallel. Write the equation of a line parallel to another line given a point it passes through. 3.7 Perpendicular Lines in the Coordinate Plane Identify the slopes of lines and determine if two lines are perpendicular. Write the equation of a line perpendicular to another line given a point it passes through. EXTRA: Finding the distance from a point to a line using slopes & equations Vocabulary for Chapter 3 3.1 Parallel Lines Perpendicular Lines Skew Lines Parallel Planes Transversal Corresponding angles Alternate exterior angles Alternate interior angles Consecutive interior angles (same side interior angles) 3.2 Two-column proof Paragraph proof Flow proof 3.6 Slope Slope-intercept form 3.7 opposites reciprocals Ohio Academic Content Standards G.CO.1 Experiment with transformations in the plane. Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length. G.CO.9 Prove geometric theorems. Apply multiple proof methods, such as narrative paragraphs, flow diagrams, coordinate proofs, two-column proofs, diagrams without words, and the use of dynamic software. Prove and apply theorems about lines and angles. Theorems include but are not restricted to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. C.GO.12 Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folder, dynamic geometric software, etc.) Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 3.1 Lines & Angles Define the following words and draw a diagram: Parallel Lines – p.129 Diagram: Perpendicular Lines – p.79 Diagram: Skew Lines – p.129 Diagram: Parallel Planes – p.129 Diagram: Transversal – p. 131 Diagram: p.129 Angle Relationships Vertical Angles: - p. 44 Linear Pair – p. 44 Corresponding Angles – p. 131 Alternate Exterior Angles – p. 131 Alternate Interior Angles – p. 131 Consecutive Interior Angles – p. 131 Practice: Q: Compare and Contrast parallel lines and skew lines. ______________________________________________________________________________ ______________________________________________________________________________ _____________________________________________________________________________________ 3.2 Types of Proof Read Example 1 on page 136 in your textbook. Notice the same proof is shown using 3 different formats. In the diagram, 𝑚∠8 = 𝑚∠5 and 𝑚∠5 = 125°. Prove that 𝑚∠7 = 55°. TWO – COLUMN PROOF: PARAGRAPH (NARRATIVE) PROOF: FLOW PROOF: 3.3 Parallel Lines & Transversals GOAL: SOLVE FOR MISSING ANGLE MEASURES Example 1: Example 2: Use properties of parallel lines to find the value of x. Tell which postulate or theorem you use. (a) (b) (c) Example 3: (d) 3.4&3.5 Proving Lines Parallel We will learn six ways to prove that lines are parallel. Transitive Property of Parallel Lines: Lines Perpendicular to a Transversal Theorem: Example 1: Is it possible to prove the lines parallel? If so, explain what theorem/postulate you used. Example 2: Find the value of x that makes line r parallel to line s. Example 3: Example 4: Determine which lines, if any, are parallel. Example 5: Example 6: Example 7: Proving the Transitive Property of Parallel Lines Example 8: You are building a bookshelf. You cut the sides, bottom, and top so that each corner is composed of two 45° angles. Prove that the top and bottom front edges of the bookshelf are parallel. 3.6 & 3.7 Slopes of Parallel and Perpendicular Lines Slope = 𝒓𝒊𝒔𝒆 ↑ ↓ 𝒓𝒖𝒏 → 𝒚𝟏 −𝒚𝟐 = 𝒙𝟏 −𝒙𝟐 Slope-Intercept Equation 𝒚 = 𝒎𝒙 + 𝒃 m = slope (0, b) = y-intercept (x, y) = any point on the line Example 1: Find the slope of the line that passes through the points (-4, -3) and (5, 3) Method 1: Use slope formula Method 2: Count rise and run Example 2: Identify the slopes of the lines given by the equations. 3 (a) 𝑦 = 4 𝑥 + 2 (b) 𝑦 − 6𝑥 = 12 (c) 4𝑦 + 3𝑥 = −12 Example 3: A slope of a line is given. Find the slopes of a parallel and perpendicular line. 2 (a) 𝑚 = 3 ∥ slope = ___________ ⊥ slope = _____________ ∥ slope = ___________ ⊥ slope = _____________ (b) 𝑚 = −6 Example 4: Find both slopes. Are the lines parallel, perpendicular, or neither? (a) (c) 1 𝑦 = −2𝑥 + 9 𝑦 = −2𝑥 − 4 (b) 𝑦 = −3𝑥 − 4 6𝑥 + 2𝑦 = 8 (d) 3.6 & 3.7 Equations of Parallel and Perpendicular Lines Slope-Intercept Equation 𝒚 = 𝒎𝒙 + 𝒃 m = slope (0, b) = y-intercept (x, y) = any point on the line ∥ Parallel lines have _______________ slopes. ⊥ Perpendicular lines have slopes that multiply to _________. The slopes are ___________________ and _______________________. ______________________________________________________________________________ Example 1: Finding equations of lines given a y-intercept. Line l has the equation 𝑦 = 𝑥 + 4. (a) Find a line parallel to line l with a y-intercept of (0, -3). (b) Find a line perpendicular to line l with a y-intercept of (0, 1). Solution: (a) First, identify the slope of the original equation. Then identify the slope of a parallel line. Use the given y-intercept and slope you found to write your equation. (b) First, identify the slope of the original equation. Then identify the slope of a perpendicular line. Use the given y-intercept and slope you found to write your equation. Check using a Graph Example 2: Finding equations of lines given a point on the line. Line l has the equation 𝑦 = −2𝑥 + 5. (a) Find a line parallel to line l that passes through the point (-4, 3). (b) Find a line perpendicular to line l that passes through the point (2, 6). Solution: (a) First, identify the slope of the original equation. Identify the slope of a parallel line. Then, use your new slope and coordinate point to solve for b. Write new equation. (b) First, identify the slope of the original equation. Identify the slope of a perpendicular line. Then, use your new slope and coordinate point to solve for b. Write new equation. Check using a Graph Distances to Lines The distance from a point to a line is the length of the perpendicular segment from the point to the line. This perpendicular segment is the shortest distance between the point and the line. For example, the distance between point A and line k is AB. Recall: 𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 Example 1: ⃡ . Find the distance from point A to 𝐵𝐷 You Try: ⃡ . Find the distance from point E to 𝐹𝐻 Example 2: Find the distance from the point (1, 0) to the line 𝑦 = −𝑥 + 3. Step 1: Find the equation of the line perpendicular to 𝑦 = −𝑥 + 3 that passes through the point (1, 0). Step 2: Use the two equations to write and solve a system of equations to find the point where the two lines intersect. Step 3: Use the distance formula to find the distance from (1, 0) to (2, 1). You Try: Find the distance from the point (6, 4) to the line 𝑦 = 𝑥 + 4. Constructing a Perpendicular Line through a Point not on the Line Constructing a Parallel Line through a Point not on the Line