4-1 NAME DATE PERIOD. Study Guide. Parallel Lines and Planes P Q, O Q. Sample answers: A J, A F, and D E

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1 4-1 NAME DATE PERIOD Pges Prllel Lines nd Plnes When plnes do not intersect, they re sid to e prllel. Also, when lines in the sme plne do not intersect, they re prllel. But when lines re not in the sme plne nd do not intersect, they re skew. Exmple: Nme the prts of the tringulr prism shown t the right. Smple nswers re given. prllel plnes: plnes PQR nd NOM prllel segments: M O nd R Q skew segments: M N nd R Q Refer to the figure in the exmple. 1. Nme two more pirs of prllel segments. Smple nswers: M N nd R P, N O nd P Q, N P nd M R, N P nd O Q, M R nd O Q 2. Nme two more segments skew to N M. P Q, O Q 3. Nme segment tht is prllel to plne MRQ. N P Nme the prts of the hexgonl prism shown t the right. 4. three segments tht re prllel to B C F E, Q P, K L 5. three segments tht re prllel to J K A B, M P, D E 6. segment tht is skew to Q P Smple nswers: A J, A F, nd D E 7. the plne tht is prllel to plne AJQ plne CDM Glencoe/McGrw-Hill T19 Geometry: Concepts nd Applictions

2 4-2 NAME DATE PERIOD Pges Prllel Lines nd Trnsversls A line tht intersects two or more lines in plne t different points is clled trnsversl. Eight ngles re formed y trnsversl nd two lines. Types of Angles Angle Definition Exmples interior lie etween the two lines 1, 2, 5, 6 lternte interior on opposite sides of the trnsversl 1 nd 6, 2 nd 5 consecutive interior on the sme side of the trnsversl 1 nd 5, 2 nd 6 exterior lie outside the two lines 3, 4, 7, 8 lternte exterior on opposite sides of the trnsversl 3 nd 7, 4 nd 8 Identify ech pir of ngles s lternte interior, lternte exterior, consecutive interior, or verticl nd nd 13 lternte exterior consecutive interior nd nd 5 lternte interior lternte exterior nd nd 16 consecutive interior verticl In the figure, A B D C nd B C A D. 7. For which pir of prllel lines re 1 nd 4 lternte interior ngles? B C nd A D 8. For which pir of prllel lines re 2 nd 3 lternte interior ngles? A B nd D C Glencoe/McGrw-Hill T20 Geometry: Concepts nd Applictions

3 4-3 NAME DATE PERIOD Pges Trnsversls nd Corresponding Angles If two prllel lines re cut y trnsversl, then the following pirs of ngles re congruent. corresponding ngles lternte interior ngles lternte exterior ngles If two prllel lines re cut y trnsversl, then consecutive interior ngles re supplementry. Exmple: In the figure m n nd p is trnsversl. If m 2 35, find the mesures of the remining ngles. Since m 2 35, m 8 35 (corresponding ngles). Since m 2 35, m 6 35 (lternte interior ngles). Since m 8 35, m 4 35 (lternte exterior ngles). m 2 m Since consecutive interior ngles re supplementry, m 5 145, which implies tht m 3, m 7, nd m 1 equl 145. In the figure t the right p q, m 1 78, nd m Find the mesure of ech ngle Find the vlues of x nd y, in ech figure. 8. (3x 5) (y 8) (6x 14) 21, 60 18, 10 10, 19 (8x 40) 5x 9y (3y 6x 10) 7y Find the vlues of x, y, nd z in ech figure (3z 18) x 72 3y (y 18) x z (y 12) 108, 36, 30 90, 93, 15 Glencoe/McGrw-Hill T21 Geometry: Concepts nd Applictions

4 4-4 NAME DATE PERIOD Pges Proving Lines Prllel Suppose two lines in plne re cut y trnsversl. With enough informtion out the ngles tht re formed, you cn decide whether the two lines re prllel. IF corresponding ngles re congruent, lternte interior ngles re congruent, lternte exterior ngles re congruent, consecutive interior ngles re supplementry, the lines re perpendiculr to the sme line, THEN the lines re prllel. Exmple: If 1 2, which lines must e prllel? Explin. AC BD ecuse pir of corresponding ngles re congruent. Find x so tht (4x 10) (4x 22) (3x 50) (2x 5) 57 (3x 9) (6x 12) 2x (6x 7) (3x 38) Given the following informtion, determine which lines, if ny, re prllel. Stte the postulte or theorem tht justifies your nswer c d, lternte exterior ngles congruent e f, lternte interior ngles congruent 9. m 7 m e f, consecutive c d, interior ngles corresponding supplementry ngles congruent e f c d Glencoe/McGrw-Hill T22 Geometry: Concepts nd Applictions

5 4-5 NAME DATE PERIOD Pges Slope To find the slope of line contining two points with coordintes (x 1, y 1 ) nd (x 2, y 2 ), use the following formul. m x y 2 2 y x 1, where x 1 x 2 1 The slope of verticl line, where x 1 x 2, is undefined. Two lines hve the sme slope if nd only if they re prllel nd nonverticl. Two nonverticl lines re perpendiculr if nd only if the product of their slopes is 1. Exmple: Find the slope of the line pssing through A(2, 5) nd B( 1, 3). Stte the slope of line prllel to. Then stte the slope of line perpendiculr to. Let (x 1, y 1 ) (2, 5) nd (x 2, y 2 ) ( 1, 3). Then m 3 ( 5) Any line in the coordinte plne prllel to hs slope 8 3. Since , the slope of line perpendiculr to the line is 3 8. Find the slope of the line pssing through the given points. 1. C( 2, 4), D(8, 12) 2. J( 4, 6), K(3, 10) 3. P(0, 12), R(12, 0) S(15, 15), T( 15, 0) 5. F(21, 12), G( 6, 4) 6. L(7, 0), M( 17, 10) Find the slope of the line prllel to the line pssing through ech pir of points. Then stte the slope of the line perpendiculr to the line pssing through ech pir of points. 7. I(9, 3), J(6, 10) 8. G( 8, 12), H(4, 1) 9. M(5, 2), T(9, 6) 7 3, , , 1 Glencoe/McGrw-Hill T23 Geometry: Concepts nd Applictions

6 4-6 NAME DATE PERIOD Pges Equtions of Lines You cn write n eqution of line if you re given the slope nd the coordintes of point on the line, or the coordintes of two points on the line. Exmple: Write the eqution in slope-intercept form of the line tht hs slope 5 nd n x-intercept of 3. Since the slope is 5, you cn sustitute 5 for m in y mx. y 5x Since the x-intercept is 3, the point (3, 0) is on the line. y 5x 0 5(3) y 0 nd x Solve for. So the eqution is y 5x 15. If you know two points on line, you will need to find the slope of the line pssing through the points nd then write the eqution. Write the eqution in slope-intercept form of the line tht stisfies the given conditions. 1. m 3, y-intercept 4 2. m 2, x-intercept 6 5 y 3x 4 y 2 5 x psses through ( 5, 10) nd (2, 4) 4. psses through (8, 6) nd ( 3, 3) y 6 7 x y x perpendiculr to the y-xis, 6. prllel to the y-xis, psses through ( 6, 4) psses through ( 7, 3) y 4 x 7 7. m 3 nd psses through ( 4, 6) 8. perpendiculr to the grph of y 4x 1 nd psses through (6, 3) y 3x 18 y 1 4 x 3 2 Glencoe/McGrw-Hill T24 Geometry: Concepts nd Applictions