Lines and Angles: Parallel Lines, Skew Lines, Perpendicular Lines, and Planes, Exercises of Analytical Geometry

Download Lines and Angles: Parallel Lines, Skew Lines, Perpendicular Lines, and Planes and more Exercises Analytical Geometry in PDF only on Docsity! Lines and Angles Lines Relationship Parallel Planes Thursday, April 9, 2020 Lines Relationship  Parallel Lines  Skew Lines  Perpendicular Lines Two lines are parallel lines if they do not intersect and are coplanar. Two lines are skew lines if they do not intersect and are not coplanar. Two lines are perpendicular lines if they intersect and form right angles. Parallel and Perpendicular Postulate Parallel Line 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. Illustration n P There is exactly one line through P parallel to line n > >• Parallel and Perpendicular Postulate Perpendicular Line 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. Illustration n P There is exactly one line through P perpendicular to line n • Skew, Parallel Lines, Planes, Perpendicular d. Plane(s) parallel to plane EFG and containing point A c. Line(s) perpendicular to CD and containing point A a. Line(s) parallel to CD and containing point A b. Line(s) skew to CD and containing point A Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure appear to fit the description? AB , HG , and EF all appear parallel to CD , but only AB contains point A. Both AG and AH is skew to CD and contain point A. BC, AD , DE , and FC all appear perpendicular to CD , but only AD contains point A. Plane ABC appears parallel to plane EFG and contains point A. Transversal Line  Transversal Line is a line that intersect two or more coplanar lines at different points l1 l2 l3 l1 l2 l3 Example: Name the transversal Line l1Transversal Line : Transversal Line : l1 l2 l3 1. 2. Angles Formed by Transversal Line  When a transversal line intersect two lines, 8 angles are formed as shown in the figure below. 1 2 3 4 5 6 7 8 l1 l2 l3 CORRESPONDING ANGLES: ALTERNATE EXTERIOR ANGLES: SAME SIDE/ CONSECUTIVE INTERIOR ANGLES: ALTERNATE INTERIOR ANGLES: 1 and 5, 1 and 8, 3 and 6, 3 and 5, 2 and 6, 3 and 7, 4 and 8 2 and 7 4 and 5 4 and 6 Angles Formed by Transversal Line Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, corresponding or vertical. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1. ) 1 and 2 2. ) 8 and 2 3. ) 5 and 4 4. ) 1 and 8 5. ) 5 and 2 6. ) 1 3and 14 7. ) 9 and 12 8. ) 9 and 14 9. ) 11 and 13 10. ) 15 and 14 corresponding vertical alternate interior alternate exterior consecutive interior consecutive interior alternate exterior corresponding vertical alternate interior Examples Theorems on Angles Formed by a Transversal If two parallel lines is cut by a transversal line then the following theorems hold.  Alternate Interior Angles are Congruent  Alternate Exterior Angles are Congruent  Consecutive Interior Angles are Supplementary  Corresponding Angles are Congruent Corresponding Angles Postulate Thursday, April 9, 2020 Theorems on Angles Formed by a Transversal Find the measure of the angle 1. ) If m1 = 1250, find m2 =_____ 2. ) If m8 =1350, find m2 =_____ 3. ) If m5 = 870, find m4 =_____ 4. ) If m1 = 1180, find m8 = _____ 5. ) If m5 = 780, find m2 =_____ 6. ) If m2 = 1260, find m4 =____ 7. ) If m9 =5x + 8 and m11 = 12x + 2 8. ) If m9 =2x + 7 and m14 = 3x - 13 9. ) If m11= 14x - 56 and m13 = 6x 10. ) If m5=5x + 25 and m2= 3x - 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 For #’s 7-10, Solve for x and the angles 1250 1350 870 1180 1020 540 x = 10 m9 = 580 m11 = 1220 x = 20 m9 = 470 m11 = 470 x = 7 m11 = 420 m13 = 420 x = 20 m6 = 1250 m2 = 550 Example Theorems on Angles Formed by a Transversal Try it yourself ! 1. If m1 = 105°, find m4, m5, and m8. Tell which postulate or theorem you use in each case. 2. If m 3 = 68° and m 8 = (2x + 4)°, what is the value of x? Show your steps. m4 = 105° m5 = 105° m8 = 105° Vertical Angles Congruence Corresponding Angles Alternate Exterior Angles x = 54 Prove Lines Parallel Theorem 3.4: Alternate Interior Angles Converse If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. Illustration In the figure, 1  2 then line a // line b Prove Lines Parallel Theorem 3.5: Alternate Exterior Angles Converse If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel. Illustration In the figure, 1  2 then line a // line b 1 2 Prove Lines Parallel Theorem 3.6: Consecutive Interior Angles Converse If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel. Illustration In the figure, m1+ m2 = 180 then line a // line b 2 2 Theorems on Parallel and Perpendicular  Three Parallel Lines Theorem If two lines are parallel to the same line then the lines are parallel to each other. Illustration If line l 1 is parallel to line l 2, Then line l 1 is parallel to l 3 line l 3 is parallel to line l 2 l 1> l 2> l 3> Theorems on Parallel and Perpendicular  Two Lines Perpendicular to a Line Theorem In a plane, If two lines are perpendicular to the same line then the lines are parallel to each other. Illustration If line l 1 is perpendicular to line l 2, l 1 l 2 line l 3 is perpendicular to line l 2 l 3 > > Then line l 1 is parallel to l 3 Theorems on Parallel and Perpendicular  Perpendicular Transversal Theorem In a plane, If a line is perpendicular to one of two parallel line then it is also perpendicular to the other line. Illustration If line l 1 is perpendicular to line l 2, l 1 l 2 line l 1 is parallel to line l 3 l 3> > Then line l 3 is perpendicular to l 2 Slope of a LINE • Example 2 • pick a starting point • do rise by going directly up or down • do run by going left or right to towards exactly at the second point m = • 5 - 4 or -5 4 Slope of the Line 2. Given Two Points (x1 , y1) and (x2 , y2) 12 12 xx yy m    Example 1. Passing through (-5, -1) and (2, -1) Solving for Slope 2. Passing through (3, -1) and (7, -5) m = -1 - 2 -1 - - 5 = 0 m = -5 - 7 -1 - 3 = -1 x1 y1 x2 y2 x1 y1 x2 y2 Slope of a LINE 3. Given an Equation: y = mx + b slope y-intercept Example: Find the slope of the line given the equation 1. y = -4x - 2 2. 3x - 2y = 6 - 2y = -3x + 6 -2 -2 -2 y = 3x - 3 2 -3x -3x m = - 4 m = 3 2 3. -2x + 5y = 10 5y = 2x + 10 5 5 5 y = 2x + 2 5 +2x +2x m = 2 5 Solving for Slope Equations of Lines in the Coordinate Plane Lesson 3.5 Slope of Parallel Lines Slope of Perpendicular Lines Writing Equation of Lines Thursday, April 9, 2020 Parallel and Perpendicular Slope  Parallel Lines has equal slope  Perpendicular Lines has negative reciprocal slope Example 1. The slope of Line1 is m = -5. Line 1 is parallel to Line2. What is the slope of Line2? 2. The slope of Line1 is m = -5. Line1 is perpendicular to Line2. What is the slope of Line2? Sample Problem 1. The slope of Line1 is m = 3. Line1 is parallel to Line2. What is the slope of Line2? 3. The slope of Line1 is m = . Line1 is perpendicular to Line2. What is the slope of Line2? 2. The slope of Line1 is m = . Line1 is parallel to Line2. What is the slope of Line2? 4. The slope of Line1 is m = . Line1 is perpendicular to Line2. What is the slope of Line2? 3 2 3 -1 3 2 Sample Problem Changing Equation Form of a Line A. Change the following equation into Slope Y-intercept Form: y = mx +b 1. y = -5 x + 2 2. 4x - 3y = 12 - 3y = -4x + 12 -3 -3 -3 y = 4 x - 4 3 -4x -4x 3. 2x – y = 2 -y = -2x + 2 y = 2 x - 2 -2x -2x Sample Problem Changing Equation Form of a Line B. Change the following equation into Standard Form Ax + By = C 1. y = 2x - 1 2. 3y + 6 = 2x -2x-2x -2x + y = -1 2x - y = 1 -2x-2x -2x + 3y + 6 = 0 - 6 - 6 -2x + 3y = - 6 2x – 3y = 6 3. y = x + 3 1 2 1 2 x 1 2 x 1 2 x + y = 3 2 x + 2y = 6 Practice Work Changing Equation Form of a Line A. Change the following equation into Slope Y-intercept Form: y = mx +b B. Change the following equation into Standard Form Ax + By = C 1. 5x - y = 2 2. 7x - 2y = 6 3. 3x + 5y = 10 4. 5x = 2 - 3y 5. 2x + y = 3 1. 5x = 2 - 3y 2. 3y = 2 – 4x 3. x = 2 + 3y 4. y = 2 – 3x 5. y = x + 2 1 3 Example: Given a slope and point, write the equation in standard form Ax + By = C and slope y-intercept form y = mx + b. Use the formula above: 1. m = 1 and (-2 , 1) x , y y = mx + b 1 = Slope Y-Int Form : y = 1x + 3 Standard Form : x – y = -3 2. m = and (-4 , 1) 3 2 x , y Slope Y-Int Form: y = + 73x 2 Standard Form : 3x – 2y = -14 STEPS 1. Look/Solve for slope m 2. Look for point (x,y) 3. Plug into y = mx + b and solve for b. 4. Write equation in two forms as required. 1-2 + b 1 = -2 + b +2+2 3 = b y = mx + b 1 = 3/2-4 + b 1 = -6 + b +6+6 7 = b Slope Intercept Formula: y = mx + b y2 – y1 x2 – x1 =m Slope Formula: Given two points, write the equation in standard form Ax + By = C and slope y- intercept form y = mx + b. Use the formula above: 3. (-2 , 5) and (-4, 6) x , y Standard Form : x + 2y = 8 Solve for slope m: -1 2 m = Slope Y-Int Form : y = x + 4 -1 2 Slope Intercept Formula: y = mx + b y2 – y1 x2 – x1 =m Slope Formula: y = mx + b 5 = -1/2 -2 + b 5 = 1 + b - 1- 1 4 = b STEPS 1. Look/Solve for slope m 2. Look for point (x,y) 3. Plug into y = mx + b and solve for b. 4. Write equation in two forms as required. 4. (5, -2) and (3 , 6) x , y Solve for slope m: -4m = y = mx + b -2 = -45 + b -2 = -20 + b +20+20 18 = b Standard Form : 4x + y = 18 Slope Y-Int Form: y = -4x + 18 Example: Writing Equation with Parallel and Perpendicular Lines 1. (-1 , 3) m = -8 x STEPS 1. Look/Solve for slope m 2. Look for point (x,y) 3. Plug-in, simplify. 4. Write equation in two forms as required. y =y -8x – 5 y = mx + b 5.) Through point (-1, 3) and parallel to 8x + y = 3 3 = -8-1+ b 3 = 8 + b -8-8 -5 = b Equation of a LINE Line r has the equation 2x + 3y = 6. Find the equation of line t perpendicular to line r that passes through the point ( -1, 2) SOLUTION Find the slope of line t To find the slope of line t, first find slope of line r Use the equation and get y by itself to find slope of line r 2x + 3y = 6 3y = -2x + 6 y = 3 2  x + 2 Slope m of line rSlope m of line t is m = 2 3 Slope of a Line  Horizontal Line H O Y Horizontal Line Zero Slope y = number is the equation Slope of a Line  Vertical Line Ver No X Vertical Line No Slope x = number is the equation “HOY VerNoX”