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Geometry Possible Scope and Sequence Curriculum Cluster 2: Parallel and Perpendicular Lines a.1 Builds on knowledge of number, operation, and quantitative reasoning; patterns, and algebraic thinking, geometry, measurement, and probability and statistics, solve meaningful problems by representing and transforming figures and analyzing relationships. a.6 Makes connections, uses multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts G.1 The student understands the structure of, and relationships within, an axiomatic system. G.2 The student analyzes geometric relationships in order to make and verify conjectures. G.3 The student applies logical reasoning to justify and prove mathematical statements . G.4 The student uses a variety of representations to describe geometric relationships and solve problems. G.7 The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. G.9 The student analyzes properties and describes relationships in geometric figures. Possible Resources TEKS G.1.A Develop awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. G.3.C Use logical reasoning to prove statements are true and find counter examples to disprove statements t are false. Instructional Scope Logic and Truth Tables: Introduction to Symbolic Logic When you answer true –false questions on a test, the basic principle of logic is applied. You know that there is only one correct answer, either true or false. The truth or falsity of a statement is called its truth value. Instructional Resources Assessment Resources Supplemental Resources Holt: Chapter 2 Introduction to Symbolic Logic, pp.128-129 Holt Lesson Tutorial Videos: CD-ROM Chapter 2 A convenient method for organizing and listing all possible combinations of truth values for a statement or compound statements is called a truth table. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 1 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.4.A Select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Compound Statements: Conjunction – p ˅ q ( read “ p and q”) is true when all of its parts are true. Union symbol - ˅ Disjunction – p ˄ q (read “ p or q”) is true if any one of its parts are true. Intersection symbol - ˄ Truth tables can be constructed for more complex compound statements involving negations ( ̴ p – read “not p”), conjunctions, and disjunctions. Example: p T T F F q T F T F Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. ̴ p F F T T ̴ p˄q F F T F Curriculum Cluster 2: Parallel and Perpendicular Lines Page 2 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Ask students to construct and complete truth tables to determine the truth value of logical and given statements for p and q. G.1.A Develop an awareness of the structure of a mathematical system, connecting definitions, postulates and theorems. G.1.B Recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes. G.1.C Compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. Non-Euclidean Geometries Euclid was the founder of plane geometry. Therefore the geometry we use with planes or about planes is called Euclidean Geometry. Euclid used postulates and deductive reasoning . There are other geometries called NonEuclidean geometries. Hyperbolic geometry is a non-Euclidean geometry. It is known for having constant curvature. This geometry satisfies all of Euclid’s postulates except the parallel postulate. Rather than having one line through a point parallel to a given line, hyperbolic geometry uses many lines parallel to the given line. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Holt : Chapter 10Spherical Geometry, pp. 726-729 High School Geometry: Supporting TEKS and TAKS (TEXTEAMS). “Bayou City Geometry,” Activities 6-9, pp. 20-26. High School Geometry: Supporting TEKS and TAKS (TEXTEAMS) “Bayou City Geometry” Reflect and Apply, p. 27. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 3 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.7.A Use one and two dimensional coordinate systems to represent points, lines, rays, line segments, and systems to represent points, lines, rays, line segments, and figures. G.9.D Analyze the characteristics of polyhedra and other 3-D figures and their component parts based on explorations and models. G.7.C Use formulas involving length, slope, and midpoint. and midpoint. Instructional Scope Euclidean Geometries are spherical geometry and taxi cab geometry. In spherical geometry the lines are like latitude and longitude of the sphere. In taxi cab geometry, distance is used. Although this geometry does use some planes, the distance is not “as the crow flies”. In taxi cab geometry, the distance between two points is how a taxi would travel to get from one point to another. There are usually many ways to get there, but must travel in horizontal or vertical directions. Taxi Cab Geometry Find the distance from A to B Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Instructional Resources High School Geometry: Supporting TEKS and TAKS (TEXTEAMS). “Spherical Geometry,” pp. 223-228. Assessment Resources Supplemental Resources Engaging Mathematics: TEKS Based Activities Geometry Non-Euclidean Geometry, pp.24-25 Geometry Clarifying Activities http://www.tenet.ed u/teks/math/clarifyin g/cageob1.html Curriculum Cluster 2: Parallel and Perpendicular Lines Page 4 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Red route: 7 blocks left 2 blocks up Total distance= 9 blocks Green route: 1 block up 5 blocks left 1 block up 2 blocks left Total distance= 9 blocks The red and green routes are only 2 of the possible routes from A to B. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 5 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Compare and contrast Euclidean and nonEuclidean (spherical, hyperbolic, and taxicab) geometries in order to illustrate the importance of precise definitions and application of postulates. For example, ask students to brainstorm about triangles. Sample answers might be: three sides, three angles, and all angles add up to 180o . Ask them to brainstorm about lines. Sample answers might be: they go on for forever; it only takes two points to make a line. Give students a tennis ball and three rubber bands (one set per group). Explain the definition of a line on a sphere is a great circle—which is a line on the surface of a sphere formed by the intersection of a plane through the center of the sphere. Have them place one rubber band around the tennis ball letting it represent a line (like an equator). Ask students what they know about perpendicular lines. See if they can place Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 6 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources another rubber band on the tennis ball that is perpendicular to the first one. Have them find the 90o angles that are formed. By placing a third rubber band on the tennis ball perpendicular to the second rubber band, the students will form a triangle, with three sides that have three 90o angles for a total of 270o . Use questioning techniques to help students draw conclusions. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 7 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.1.A Develop an awareness of the structure of a mathematical system, connecting definitions, postulates and theorems. G.5.B Use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. G.9.A Formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models. Instructional Scope Transversals and Special Pairs of Angles Key Vocabulary: transversal, exterior angles, interior angles, corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles A line that intersects two other lines at different points is called a transversal. Two lines and a transversal form eight angles. Some pairs of these angles have special names. Instructional Resources Holt Geometry – Chapter 3 pp. 146-211 Resources for All Learners Holt Resource Book: Chapter 3 Developing LearnersPractice A, Reteach, Questioning Strategies, Modified Chapter 3 Resources On-Level LearnersPractice B, Multiple Representations, Cognitive Strategies In the figure, t is the transversal that intersects lines m and n. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Advanced LearnersPractice C, Challenge, Critical Thinking Assessment Resources Instructional Fair Geometry 8764(IF0280) pp.32-35. Holt: Chapter 3 Test & Practice Generator – OneStop Planner Chapter Test (Levels A, B, C) Holt Assessment Resources, pp. 47-58 Supplemental Resources Holt 3.1 -3.6 Power-Point Presentations CD-ROM Holt 3.1-3.6 Lesson Tutorial Videos Holt: 3.1-3.6 One Stop Planner, CD-ROM, Disc 1 Holt Texas Lab Manual: Geometry Lab Chapter 3 www.kutasoftware. com/freeige.html Parallel Lines and Transversals Curriculum Cluster 2: Parallel and Perpendicular Lines Page 8 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Holt Exterior angles are angles on the outside of the two lines. In the figure, angles 1, 2, 7, and 8 are exterior angles. Technology Lab:3.2 Explore Parallel and Perpendicular Lines, p. 154 online.dpsk12.org./ math-resources / Interior angles are angles that are inside the two lines. In the figure, angles 3, 4, 5, and 6 are interior angles. Corresponding angles are angles in the same positions on the two lines. In the figure, < 1 and < 5 are corresponding angles. The other pairs of corresponding angles are < 2 and < 6, <4 and < 8, and <3 and <7. Alternate exterior angles are angles outside the two lines and on opposite sides of the transversal. < 1 and < 7 are alternate exterior angles. < 2 and < 8 are alternate exterior angles. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 9 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Consecutive interior angles are angles inside the two lines and on the same side of the transversal. < 3 and < 6 are consecutive interior angles. < 4 and < 5 are consecutive interior angles. Example: Identify the lines and transversal that form the pair of angles. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 10 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources < 9 and < 13 < 9 is formed by lines q and n. < 13 is formed by lines q and l. Therefore, the two lines are n and l, and the transversal is q. Example: Identify which pair of special angles < 3 and < 9 are. < 3 and < 9 are inside lines p and q, and on opposite sides of transversal n. Therefore, < 3 and < 9 are alternate interior angles. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 11 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Parallel Lines and Transversals: Visual Activity: Have students use lined paper to draw two parallel lines and a transversal that is not perpendicular to the lines. Instruct students to shade the acute angles one color and the obtuse angles another color. Let students use a protractor to see that all the angles shaded the same color are congruent and that pairs of angles shaded different colors are supplementary/ Holt Manipulative Kit: protractors patty paper Kinesthetic Activity: Have students draw a pair of parallel lines and a transversal on patty paper. Tear the paper between the parallel lines and overlay the two parts to show that the angles are congruent. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 12 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.1.A Develop an awareness of the structure of a mathematical system, connecting definitions, postulates and theorems. Instructional Scope Instructional Resources ANGLES FORMED BY PARALLEL LINES Holt 3.3- Proving Lines Parallel, pp. 162-169 KEY VOCABULARY: Parallel lines, skew lines Two lines in the same plane that do not intersect are parallel lines. Two lines that are not in the same plane, but do not intersect , are skew lines. G.2.A Use constructions to explore attributes of geometric figures and make conjectures about geometric relationships. When using parallel lines, the symbol on a figure is a larger bolder arrow, or the larger arrows in a different color. In writing, the symbol is ||. Assessment Resources Supplemental Resources Engaging Mathematics: TEKS Based Activities Geometry Angle Relationships, Activity 2, p. 36 Angle Puzzler, p.37 Angle Relationships, p. 38 Parallel Universe, p. 39 G.3.C Use logical reasoning to prove statements are true and find counterexamples to disprove statements that are false. In the figure, m ||n. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 13 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources We know that two lines intersected by a transversal form special pairs of angles. When these lines are parallel, the special pairs of angles have special relationships. Corresponding Angles Postulate If two lines are parallel and cut by a transversal, then each pair of corresponding angles are congruent. 1 2 4 3 Geometry Lab:3.3 Constructing Parallel Lines, p.170 5 6 8 7 Alternate Interior Angles Theorem If two lines are parallel and cut by a transversal, then each pair of alternate interior angles are congruent. 3 5 4 6 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 14 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Alternate Exterior Angles Theorem If two lines are parallel and cut by a transversal, then each pair of alternate exterior angles are congruent. 1 7 2 8 Consecutive Interior Angles Theorem If two lines are parallel and cut by a transversal, then each pair of consecutive interior angles are supplementary. 4 and 5 are supplementary. 3 and 6 are supplementary. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 15 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Example: Given m ||n. If m 1 = 120°, find m 8. The m 5 = 120 because 1 and 5 are corresponding angles. m 5 + m 8 = 180°, since they form a linear pair. 120 + m 8 = 180 m 8 = 60°. Example: Using the figure above, if m 4 =80°, find the m 7. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 16 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources 4 and 5 are supplementary because they are consecutive interior angles. m 4 + m 5 = 180° 80 + m 5 = 180 m 5 = 100°. Therefore, m 7 = 100° because 5 and 7 are vertical angles, and vertical angles are congruent. Example: Using the figure above, if m 3 =5x – 9 and m 5 = 3x + 3, find the value of x. 3 and 5 are alternate interior angles, so they are congruent. m 3 = m5 5x – 9 = 3x + 3 2x – 9 = 3 2x = 12 x=6 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 17 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Identifying or Proving Parallel Lines G.9.A Formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models. Sometimes we have to determine if lines are parallel. To do this, we need to use the angle relationships and the converses of the Parallel Lines theorems and postulate. Corresponding Angles Converse If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. Alternate Interior Angles Converse If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. Alternate Exterior Angles Converse If two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 18 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Consecutive Interior Angles Converse If two lines are cut by a transversal and consecutive interior angles are supplementary, then the lines are parallel. Example: Use the given information to determine if lines m and n are parallel. p 1 2 3 m 4 5 6 n 7 8 1 5 m || n Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 19 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources The angles are corresponding angles, and by the Corresponding Angles Converse, the lines are parallel. 3 6 m || n The angles are alternate interior angles, and by the Alternate Interior Angles Converse, the lines are parallel. 2 supplementary to 6 m is not parallel to n The angles are corresponding. In order for the lines to be parallel, corresponding angles must be congruent, not supplementary. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 20 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Perpendicular Lines G.2.A Use constructions to explore attributes of geometric figures and make conjectures about geometric relationships. G.3.E Use deductive reasoning to prove a statement. Key vocabulary: Perpendicular Lines Two lines are perpendicular if they intersect to form right angles. Rays and segments are parts of lines, and can also be perpendicular. Holt 3.4 – Perpendicular Lines, pp. 172178 AB CD G.9.A Formulate and test conjectures about properties of parallel and perpendicular lines based on explorations and concrete models. C A B D Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 21 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources XP XC P X C Holt Geometry Lab:3.4Constructing Perpendicular Lines, p. 179 Example: If m XQY = 2x and m YQZ = 4x + 6, find the value of x. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 22 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Since the angles have a sum of 90 degrees, 2x + 4x + 6 = 90 6x + 6 = 90 6x = 84 X = 14 Engaging Mathematics: TEKS Based Activities Geometry (Slope of a Line) Time to Go Home, p. 31 Slopes of Lines Key Vocabulary: slope G.7.A Use one- and two-dimensional coordinate system to represent points, lines, rays, line segments, and figures. The slope of a line is the ratio of vertical change to horizontal change. The only exception to this is the vertical line, which has an undefined slope. The slope m = change in y = rise change in x run Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Holt 3.5 – Slopes of LinesH pp. 182-187 One and TwoDimensional Coordinate System, Activity 3, p. 32 Which Line is It, Anyway? p. 33 Curriculum Cluster 2: Parallel and Perpendicular Lines Page 23 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.7.B Use slopes and equations of lines to investigate geometric relationships. Instructional Scope Instructional Resources Assessment Resources Supplemental Resources or m = y2 – y1 x2 – x1 Example: Find the slope of the line. G.7.C Derive and use formulas involving length, slope, and midpoint. Determine the points on the graph. The points are ( 1,1) and ( -1, -2). m = -2 - 1 -1 – 1 m = -3 -2 m=3 2 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 24 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Notice the line rises left to right, and the slope is positive. Similarly, when a line has a negative slope, it will fall left to right. Instructional Resources Assessment Resources Supplemental Resources www.kutasoftware. com/freeige.html Parallel Lines in the Coordinate Plane Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals. Example: Determine the slope of a line parallel to the line which contains points ( -5, 9) and ( -4, 7). Find the slope of the line containing these points. m=7-9 -4 – (-5) m = -2 1 m = -2 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 25 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources The slope of this line is - 2. Parallel lines have equal slopes. Therefore, any line parallel to the line with slope -2 will also have slope of -2. Example: Determine the slope of any line perpendicular to the line which contains the points ( 2, 0) and ( 4, -6). Find the slope of the line containing the points. m = -6 – 0 4–2 m = -6 2 m = -3 The slope of this line is -3. Perpendicular lines have slopes that are opposite reciprocals. Therefore, any line that is perpendicular to the line with slope -3 will have a slope of 1 . 3 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 26 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.7.B Use slopes and equations of lines to investigate geometric relationships, including parallel lines perpendicular lines, and special segments of triangles and other polygons. Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Investigating Slopes A graphing calculator can help in exploring graphs of parallel and perpendicular lines. To graph a line on a calculator, enter the equation of the line in slope-intercept form. The slope intercept form of the equation is y = mx + b, where m is the slope and b is the y-intercept. Example: y = 2x + 3 has a slope of 2 and crosses the yaxis at (0,3). Holt Technology Lab 3.6: pp. 188-189 Activity: Have students use the table feature to make a table of values for y = 3x -4 and y = 3x + 1. Use the table to find points on each line, and calculate the slopes to verify that the lines are parallel. Then make a table of values for y = 3x – 4 and y = - 1/3 x , find two points, and calculate the slopes to verify that the lines are perpendicular. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 27 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.7A Use one and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.7B Use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. G. 7C Derive and use formulas involving length, slope, and midpoint. Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Writing Equations of Lines Key Vocabulary: slope- intercept form Linear equations can be written in several forms. The most common is the slopeintercept form, y = mx + b, where m is the slope and b is the y intercept. y – y1 = m ( x – x1) is called the point- slope form of the equation. We will need to use this equation to input our values, then solve the equation for y to get the slope-intercept equation. Holt 3.6 – Lines in the Coordinate Plane, pp. 190197 tx.geometryonline .com Example: Write the equation of the line that has slope 2, and goes through the point ( -3, 4 ). Using the slope, m= 2, and the point ( -3, 4 ) as ( x1, y1 ), we will write the equation in pointslope form. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 28 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources y – 4 = 2 ( x – ( -3) ) y – 4 = 2 ( x + 3) Solving the equation for y, y – 4 = 2x + 6 y = 2x + 10 Example: Write the equation of the line that contains the points ( 0, -6 ) and ( -4 , - 3). We need to find the slope of the line. m= 6 (3) 0 (4) m= 3 4 Use this slope with one of the points. It does not matter which one. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Geometry Assessments: The Charles A. Dana Center, Cross Country Cable, pp. 5-12 Whitebeard’s Treasure, pp. 13-18 Curriculum Cluster 2: Parallel and Perpendicular Lines Page 29 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources 3 and the 4 point ( 0, -6) in the point- slope equation. We will use the slope y – (-6 ) = y+6= y= G.7A Use one and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. G.7B Use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. 3 ( x – 0) 4 3 x 4 3 x -6 4 Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals. Standard – Form of a Linear Equation in Two Variables Ax + By = C, where A and B are not both zero. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 30 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G. 7CDerive and use formulas involving length, slope, and midpoint. Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Write the standard-form of the equation in slope- intercept form ( solve for y): y = -A/B x + C/B m = -A/B (slope) and b = C/B (y-intercept) So a quick way to check whether lines have the same slope is to find –A/B for both lines. If the slopes are the same, then find C/B to see if the lines coincide(same slope and same y-intercept). Example: Write the equation of the line that goes through ( 3, -4) and is parallel to the line 2x – 3y =6. Find the slope of the line. To do this, we solve for y, the slope intercept form of the equation. 2x – 3y = 6 - 3y = - 2x + 6 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 31 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope y= Instructional Resources Assessment Resources Supplemental Resources 2 x -2 3 The slope of the line is 2 . 3 To find the equation of a line that is parallel to this line, we will use the same slope, and the given point. y – ( -4) = y+4= y= 2 (x–3) 3 2 x -2 3 2 x -6 3 This is the equation of the line through the given point, and parallel to the given line. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 32 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Instructional Resources Assessment Resources Supplemental Resources Example: Write the equation of the line that contains the point ( 1, -5) and is perpendicular to the line with the equation -3x + 4y = 12. We find the slope of the given line. -3x + 4y = 12 4y = 3x + 12 3 y= x+3 4 3 . The slope of a line 4 perpendicular to it will be the opposite 3 reciprocal of . The new slope to use will be 4 4 . 3 The slope of this line is The equation of the line that goes through 4 ( 1, -5 ) and has slope is 3 Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 33 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope y – ( -5) = y+5= y= Instructional Resources Assessment Resources Supplemental Resources 4 ( x – 1) 3 4 4 x + 3 3 4 11 x 3 3 This line is perpendicular to the line whose equation is – 3x + 4y = 12. Equations of Vertical and Horizontal Lines Vertical Line: x = a real number The slope of a vertical line is undefined (no slope). Horizontal Line: y = a a real number The slope of a horizontal line is zero. Slope- Intercept Form: y = 0x + b ( horizontal line) Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Curriculum Cluster 2: Parallel and Perpendicular Lines Page 34 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS G.7B Use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. Instructional Scope Activity: Ask students to draw a pair of parallel lines, a pair of intersecting lines, and than write a system of equations for each pair. Some students may prefer solving systems of equations by other methods, such as graphing or substitution. Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Instructional Resources Assessment Resources Supplemental Resources Holt: Systems of Equations – Algebra Review, pp. 152-153 Curriculum Cluster 2: Parallel and Perpendicular Lines Page 35 of 36 Geometry Possible Scope and Sequence Possible Resources TEKS Instructional Scope Possible Scope and Sequence, Geometry © 2005 Region 4 Education Service Center. All rights reserved. Instructional Resources Assessment Resources Supplemental Resources Curriculum Cluster 2: Parallel and Perpendicular Lines Page 36 of 36