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3-1 Lines and Angles Parallel Lines – Lines that are coplanar & do not intersect. __ __ __ __ BF II EJ II DH II CG Parallel Planes – Planes that do not intersect. BFG II EJH Perpendicular Lines – lines that intersect at a 90 degree angle. BF I FJ Skew Lines – lines that are not coplanar, not parallel, and do not intersect. Line CG is skew to Line JH Example A: Refer to the figure. 1) Name all planes that are parallel to plane DEH. CFG 2) Name all segments that are parallel to AB . DC II HG II EF 3) Name all segments that intersect with GH . HD, HE, GC, GF 4) Name all segments that are skew to CD . Lines FG, EH, BF, AE 5) Name a pair of parallel segments. SR & NM 6) Name a pair of skew segments MR & KL 7) Name a pair of perpendicular segments KL & NK 8) Name a pair of parallel planes RSN & QPK Transversal – A line that intersects two coplanar lines at two different points. Transversal t and the other two lines r and s form 8 angles. Alternate Interior Angles – nonadjacent angles that lie on opposite sides of the transversal, and INSIDE (BETWEEN) the other two lines. <2 & <7; <3 & <6 Alternate Exterior Angles – nonadjacent angles that lie on opposite sides of the transversal, and OUTSIDE the other two lines. <1 & <8; <4 & <5 Corresponding Angles – lie on the same side of the transversal, and on the same sides of the other two lines Consecutive (Same Side) Interior Angles – lie on the same side of the transversal and INSIDE (or between) the other two lines. Consecutive (Same Side) Exterior Angles – lie on the same side of the transversal and OUTSIDE the other two lines. Example B: Use the figure to identify each pair of angles. 1) <1 and <5 Corresponding Angles 2) <9 and <15 Alt. Exterior Angles 3) <2 and <8 Alt. Interior Angles 4) <5 and <15 Alt. Exterior Angles 5) <10 and <14 Corresponding Angles 7) <4 and <7 Same Side Exterior 6) <3 and <10 Same Side Interior 8) <4 and <10 Alt. Interior Angles 3-2 Angles Formed by Parallel Lines and Transversals When two parallel lines are cut by a transversal, the following pairs of angles are congruent. Corresponding angles Alternate Interior angles Alternate Exterior angles When two parallel lines are cut by a transversal, the following pairs of angles are supplementary. Consecutive Exterior angles Consecutive Interior angles Theorems and Postulates You Must Know: Alternate Interior Angle Theorem: II --- Alt. Int. <’s are congruent Alternate Exterior Angle Theorem: II --- Alt. Ext. <’s are congruent Corresponding Angle Postulate: II --- Corresponding <’s are supplementary. Consecutive Interior Angle Theorem: II --- Same Side Int. <’s are supplementary. Consecutive Exterior Angle Theorem: II --- Same Side Ext. <’s are supplementary. Example A: 1) Name the 7 angles that are congruent to <1 . 1 2 5 6 3 4 7 8 3, 6, 8, 9, 11, 14, 15 2) Name the 8 angles that are supplementary to < 11 . 9 10 13 14 11 12 15 16 12, 15, 7, 4, 2, 5, 10, 13 Example B: In the figure, m<3=110, m<10=95. Find the measure of the missing angles. 1) <1 85 2) <13 95 3) <6 85 4) <16 110 5) <2 95 6) <15 70 1 2 5 6 9 10 13 14 3 4 7 8 11 12 15 16 Example C: Find x and y from the figures. 2) 1) 8y+2 10x 3y+1 25y-20 4x-5 3x+11 4x-5 = 3x+11 3y +1 + 4x-5 = 180 10x + 8(6) +2 =180 8y +2 + 25y-20 = 180 x-5 =11 3y +1 +4(16)-5 =180 10x=130 33y -18 =180 x =16 3y +60=180 3y = 120 x =130 y=40 33y =198 y=6 Example C: Find x and the missing angles from the figures. 1) 2) 2x-135=75 2x=210 x=105 x= 105 2x+10=3x-15 10=x-15 25=x M<ABD=2(25)+10 =60 M<CBD=120 m< ABD = 75 4) 3) Find x and y in the figure. 5x + 4y =55 5x +4(5)=55 25x + 4y =120 -(5x +5y=60) 5x=35 -(25x +5y=125) Y=5 x=7 Y=5 25x +5(5)=125 25x=100 x=4 3-3 Proving Lines Parallel More Theorems and Postulates You Must know: Remember what the Converse is??? Converse Alternate Interior Angle Theorem: Alt. Interior angles are congruent ------- II Converse Alternate Exterior Angle Theorem: Alt. Exterior angles are congruent ------- II Converse Corresponding Angle Postulate: Corresponding angles are congruent ------- II Converse Consecutive Interior Angle Theorem: Same Side Int. angles are supplementary ---- II Converse Consecutive Exterior Angle Theorem: Same Side Ext. angles are supplementary -- II Exercises: Determine which lines, if any, are parallel. State the postulate or theorem to justify your answer. SSI j II k Corresponding j II k Alt. Ext. l II m 11x -25=7x+35 4x – 25 = 35 4x=60 X=15 X=21 Find the values of x and y. 5x+90=180 4) 5x=90 x=18 6y + 3y=180 5) x=60 9y=180 Y + 90=180 y=20 Y=90 7) 6) y + 140 =180 y = 40 2x + 10 =40 2x =30 x=15 Y + 70 + 40=180 y=70 X=70 Find the values of x, y, and z. 8) 9) x= y= z= x= y= z= 10 ) 11) x= x= y= y= z= z= 3-4 Perpendicular Lines Perpendicular Bisector – The shortest segment from a to a segment from the is the to the . Exercises: Draw the segment that represents the distance indicated. to 1) ABF , E to BF to 2) AHJ A E B , J to AH A F to 3) ADC to , ABC B C H J to 4) TVW , S to VW T S A W D E V Exercises: In the figure below, BH AE CF, AE BH, BC BC, CF GD, CE. Name the segment whose length represents the distance between the following points and lines. C B 5) B AE to 6) G 7) C BH to D CE to A H G F E 8) F to BC l m n p q r s t Exercises: Find x and y. C 7x-12 (7y-1)° B 2x + 13 D Three new theorems to remember: 9) 10) A Prove: p r 3-5 Slope of Lines Slope/Rate of Change – Example A: Find the slope of the line that passes through the points. 1) ( 4 ,5) and (2, 1) m = 2) (5,1) and (7, 3) m= 3) ( 3 ,2) and (3,2) m= 4) ( 8 ,9)and ( 8 , 8) m = Positive Slope – Negative Slope – Zero Slope – Undefined – Parallel Lines – Perpendicular Lines – Example B: Determine whether GH and RS are parallel, perpendicular, or neither. 1) G(14,13), H(-11,0), R(-3,7), S(-4,-5) 2) G(15,-9), H(9,-9), R(-4,-1), S(3,-1) 3) G(-4,-10), H(8,14), R(0,4), S(16,-4) Example C) Find the slope of each line. 1) AB = 6) parallel to AC = 2) AC = 7) perpendicular to AD= 3) AD= 8) perpendicular to AB = 4) CD= 5) CD= 9) perpendicular to AC = 10) perpendicular to CD= 3-6 Lines in the Coordinate Plane Slope Intercept Form – m= b= Vertical Line Horizontal Line Example A Write the equation of each line with the given information in slope intercept form. 1) Line with slope of 3 through (2,1) 2) line through (0,4) and (-1,2) 3) line with x-intercept 2 and y-intercept 3 4) line through (-3,2) and (1,2) Example B: Graph the line passing through the given point with the given slope and write each in slope intercept form. 1) (0,4), m 2) ( 3 ,0), m 0 3) (1,4), undefined Example C: Graph the line that satisfies the conditions. 1) passes through ( 2 ,2), parallel to a line whose slope is 2) passes through (0,1), perpendicular to a line whose slope is Example D: Graph each line. 1) y 2x 3 2) y 4 3) y x Example E: Decide whether the lines are parallel, perpendicular, coincide, or none. 1) y = 3x + 7 y = –3x – 4 2) Example F: Write the equation of each line. b Line b: m= b= Line m : m= b= Line r: m= b= Line p: m= b= m r p