Transcript Document
Chapter 3 Notes 3.1 Lines and Angles Two lines are PARALLEL if they are COPLANAR and do not INTERSECT Two lines are SKEW if they are NOT COPLANAR and do not INTERSECT Arrows on line mean they are parallel B F A C E H D G Two planes that do not Intersect are called PARALLEL planes. A line and a plane are parallel if they do not intersect. B F A C E H D G Line segments and rays can be parallel too! O R Y T O R Y T As long as the lines going through them are also parallel. R T O Y O R T Y Let’s name some parallel planes, lines, and some skew lines. B F A C E H D G Parallel line postulate If there is a line and a point not on the line, there is EXACTLY one parallel line through the given point. Perpendicular line postulate If there is a line and a point not on the line, there is EXACTLY one perpendicular line through the given point. Not Parallel A ______________ is a line that INTERSECTS two or more COPLANAR lines at different points. Two angles are ________________ ANGLES if they occupy ________________ positions. 1 2 3 4 Two angles are ________________ __________________ if they LIE 5 6 _________ the two lines on 7 8 __________ sides of the ________________. Two angles are _____________________________ if they LIE __________ the two lines on ___________ sides of the TRANSVERSAL. Two angles are ______________________________ (also called same side interior angles) if they LIE _________ the two lines on the _______ sides of the TRANSVERSAL. Given a point off a line, draw a line perpendicular to line from given point. 1) From the given point, pick any arc and mark the circle left and right. 2) Those two marks are your endpoints, and construct a perpendicular bisector just like the previous slide. 3.3 – Parallel Lines and Transversals Corresponding Angles Postulate (CAP) If two lines cut by transversal are ||, then the corresponding angles are congruent If m || n, then 1 2 m Note, need to use def ' s to change to m n 1 2 Alternate Interior Angles Theorem (AIA Thrm) m If two lines cut by transversal are ||, then the alternate interior angles are congruent n Consecutive Interior Angles Theorem (CIA Thrm) If two lines cut by transversal are ||, then the consecutive interior angles are supplementary 1 2 3 4 5 6 Alternate Exterior Angles Theorem (AEA Thrm) If two lines cut by transversal are ||, then the alternate exterior angles are congruent Perpendicular Transversal If a transversal is perpendicular to one of two || lines, then it is perpendicular to the other. If m || n, m t then, n t m 1 n 2 t Find the measure of angles 1 – 7 given the information below. 1 2 3 4 5 7 6 800 12x 20 Find x, y 14x 4 4y Find x, y, and the measure of all angles m1 6 y m3 4 y 32 m2 5 x 4 3 2 1 m4 3 x 10 Find w, x, y, z, and the measure of all angles m2 5 x 24 m3 y 1 4 2 m1 8 x m4 z 20 m5 w 3 5 3.4 – Proving Lines are Parallel Simply stated, the postulates and theorems yesterday have TRUE converses Converse of Corres Post. Converse of Alt. Int. Thrm. If the alt int ' s are , If the corres ' s are , then the lines are || then the lines are || Conv AIA Thrm Conv CAP If 1 3 then n || p m 1 If 2 3 then n || p n 2 3 5 p Converse of Consecutiv e Converse of Alternate Exterior Thrm If the Alternate exterior ' s are congruent, then the lines are || Conv AEA Thrm Interior Thrm If the consecutiv e int ' s are supp, then the lines are || Conv CIA Thrm If 3, 4 sup, then n || p If 1 5, then n || p m 1 4 n 2 3 5 p I show the angles, you say what theorem makes the lines parallel. m 1,5 congruent 3,6 congruent 3,5 supplementary 1 2 3 4 5 6 7 8 n p 1,8 congruent 4, 8 congruent 3, 5 congruent 5, 8 congruent Which lines are parallel? A 35 40 B 38 35 D C You try it! Are l and m parallel? How? l l m 30o 40o m 110o 44o 60o 66o Which lines are parallel? How? l m p n 40o 80o discuss 50o 80o You try it! What does x have to be for l and m to be parallel? l m (x + 40)o l 70o xo m (3x)o m Proving Conv AEA Thrm (Can' t use Conv AEA, use conv CAP) Given : 1 8 Prove : n || p 1 2 3 4 5 6 7 8 n p Given : 2 and 8 are supplement ary Prove : n || p 3.5 – Using Properties of Parallel Lines Copy an angle. 1) Draw a ray 2) Use original vertex, make radius. 3) Transfer radius to the ray you drew, and draw an arc. 4) Set radius from D and E, and transfer it to the new lines, setting the point on F and draw an intersection on the arc, then connect the dots. Given a line and a point, construct a line parallel to the given line through the given point. 1) Pick any point on the line, draw a line from there through the given point. 2) Using the angle formed by the given line and the drawn line, make a congruent angle using the given point as the vertex. 3.6 – Parallel Lines in the Coordinate Plane SLOPE FORMULA!! MEMORIZE!! SLOPE = m = y2 – y1 x2 – x1 Find points (1, 0) (4, -1) and label x1 y1 x2 y2 Plug into formula y Reduce Fraction x SLOPE = m = SLOPE FORMULA!! MEMORIZE!! SLOPE = m = y2 – y1 Find points (-2, -1) (2, 5) and label x1 y1 x2 y2 x2 – x1 Plug into formula Reduce Fraction y x SLOPE = m = Postulate: Slopes of Parallel Lines In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope. Any two vertical lines are parallel. Basically Same slope means parallel. Find the slope between each set of points. See which ones match up to be parallel. (4, 3) (2, 0) (2, 3) (-5, 2) (-1, 3) (1, 2) (-2, -1) (-1, 3) (-2, -1) (-1, -2) (-3, 0) (-8, -4) y mx b y y1 m( x x1 ) Ax By C Slope-intercept form Point-slope form Standard form Write the equation of the line given a point and a slope in SLOPE-INTERCEPT FORM 1 (4,2) m 2 x1 y1 Point Slope Form y y1 m( x x1 ) Slope - Intercept Form y mx b Write the equation of the line parallel to y 4x 2 and going through (8,3) in slope - intercept form 4 Write the equation of the line parallel y x - 5 5 with a y - intercept of 2 in slope - intercept form y y1 m( x x1 ) Write the equation of the 1 line parallel to y x 2 3 and going through (-3,1) in slope - intercept form y mx b Write the equation of the line parallel to y 2 x - 3 with y - intercept - 5 in slope - intercept form y y x x y y1 m( x x1 ) Write the equation of the 4 line parallel to y x 5 3 and going through (6,-1) in slope - intercept form y mx b Write the equation of the line parallel to 2 y x 1 and going through 5 (10,-4) in slope - intercept form Grade of a road, it’s rise over run, then changed into a percent. 2% grade 2 100 3.7 – Perpendicular Lines in the Coordinate Plane Solve for y, change it to ‘y =‘ 2 y 7 ( 2 x 6) 3 y 2 3( x 1) Distribute Get y by itself Notice how by solving for y, we put it in slope intercept form, now we can find the slope. Parallel and Perpendicular Lines Parallel Lines have the ___________ slope Blue Green What do you notice about the lines and the slope? Slopes are opposite reciprocals, or slopes multiply to equal -1 Also, vertical and horizontal lines are perpendicular Parallel Lines, SAME SLOPE Perpendicular Lines, opposite reciprocal. State the slopes of the line parallel and perpendicular to the slopes on the left. Slope Parallel 2 2 5 7 4 Perpendicular Find the slope between each set of points. See which ones match up to be perpendicular. (4, 3) (2, 0) (2, 3) (-5, 3) (-1, 3) (3, 2) (-2, -1) (-1, 3) (-2, -1) (1, -2) (-3, 0) (0, 4) Find the slope of each line, then pair up the perpendicular and parallel lines. 2 y x3 3 2x 3y 6 3x 2 y 0 3 x 2 y 12 Write the equation of the line PERPENDICU LAR to y 4x 2 and going through (8,3) in slope - intercept form 2 Write the equation of the line PERPENDICU LAR to y x 3 and going through (-2,1) in slope - intercept form y mx b Write the equation of the line y y1 m( x x1 ) Write the equation of the line PERPENDICU LAR to 1 y x 2 and going through 3 (-3,1) in slope - intercept form PERPENDICU LAR to y 2 x - 3 and going through (-4,1) in slope - intercept form y y x x y y1 m( x x1 ) Write the equation of the line PERPENDICU LAR to 4 y x 5and going through 3 (8,-1) in slope - intercept form y mx b Write the equation of the line PERPENDICU LAR to 2 y x 1 and going through 5 (10,-4) in slope - intercept form