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4-1 NAME ______________________________________DATE __________PERIOD______ Study Guide Student Edition Pages 142–147 Parallel Lines and Planes When planes do not intersect, they are said to be parallel. Also, when lines in the same plane do not intersect, they are parallel. But when lines are not in the same plane and do not intersect, they are skew. Example: Name the parts of the triangular prism shown at the right. Sample answers are given. parallel planes: planes PQR and NOM 苶 and R 苶Q 苶 parallel segments: 苶 MO skew segments: M 苶N 苶 and R 苶Q 苶 Refer to the figure in the example. 1. Name two more pairs of parallel segments. Sample answers: M 苶N 苶 and R 苶P 苶, 苶 NO 苶 and P 苶Q 苶, 苶 NP 苶 and M 苶R 苶, N 苶P 苶 and O 苶Q 苶, 苶 MR 苶 and O 苶Q 苶 2. Name two more segments skew to 苶 NM 苶. 苶Q P 苶, O 苶Q 苶 3. Name a segment that is parallel to plane MRQ. N 苶P 苶 Name the parts of the hexagonal prism shown at the right. 苶C 苶 F 苶E 苶, Q 苶P 苶, K 苶L 苶 4. three segments that are parallel to B 5. three segments that are parallel to J 苶K 苶 A 苶B 苶, M 苶P 苶, D 苶E 苶 6. a segment that is skew to 苶 QP 苶 Sample answers: A 苶J 苶, A 苶F 苶, and D 苶E 苶 7. the plane that is parallel to plane AJQ © Glencoe/McGraw-Hill plane CDM T19 Geometry: Concepts and Applications 4-2 NAME ______________________________________DATE __________PERIOD______ Study Guide Student Edition Pages 148–153 Parallel Lines and Transversals A line that intersects two or more lines in a plane at different points is called a transversal. Eight angles are formed by a transversal and two lines. Types of Angles Angle Definition Examples interior lie between the two lines ⬔1, ⬔2, ⬔5, ⬔6 alternate interior on opposite sides of the transversal ⬔1 and ⬔6, ⬔2 and ⬔5 consecutive interior on the same side of the transversal ⬔1 and ⬔5, ⬔2 and ⬔6 exterior lie outside the two lines ⬔3, ⬔4, ⬔7, ⬔8 alternate exterior on opposite sides of the transversal ⬔3 and ⬔7, ⬔4 and ⬔8 Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, or vertical. 1. ⬔6 and ⬔10 2. ⬔14 and ⬔13 3. ⬔14 and ⬔6 4. ⬔1 and ⬔5 5. ⬔12 and ⬔15 6. ⬔2 and ⬔16 alternate exterior alternate interior consecutive interior consecutive interior alternate exterior vertical In the figure, 苶 AB 苶 苶 苶 and 苶 BC 苶 苶 苶. DC AD 7. For which pair of parallel lines are ⬔1 and ⬔4 alternate interior angles? B 苶C 苶 and A 苶D 苶 8. For which pair of parallel lines are ⬔2 and ⬔3 alternate interior angles? A 苶B 苶 and D 苶C 苶 © Glencoe/McGraw-Hill T20 Geometry: Concepts and Applications NAME ______________________________________DATE __________PERIOD______ 4-3 Study Guide Student Edition Pages 156–161 Transversals and Corresponding Angles If two parallel lines are cut by a transversal, then the following pairs of angles are congruent. corresponding angles alternate interior angles alternate exterior angles If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. In the figure m 储 n and p is a transversal. If m⬔2 ⫽ 35, find the measures of the remaining angles. Example: Since m⬔2 ⫽ 35, m⬔8 ⫽ 35 (corresponding angles). Since m⬔2 ⫽ 35, m⬔6 ⫽ 35 (alternate interior angles). Since m⬔8 ⫽ 35, m⬔4 ⫽ 35 (alternate exterior angles). m⬔2 ⫹ m⬔5 ⫽ 180. Since consecutive interior angles are supplementary, m⬔5 ⫽ 145, which implies that m⬔3, m⬔7, and m⬔1 equal 145. In the figure at the right p q, m⬔1 ⴝ 78, and m⬔2 ⴝ 47. Find the measure of each angle. 1. ⬔3 2. ⬔4 3. ⬔5 4. ⬔6 5. ⬔7 6. ⬔8 102 102 133 78 47 7. ⬔9 47 133 Find the values of x and y, in each figure. 8. 9. (3x 5) ° (y 8)° (6x 14)° 10. 6x ° 5x ° 9y ° 21, 60 (8x 40)° 18, 10 7y ° (3y 10)° 10, 19 Find the values of x, y, and z in each figure. 11. 12. (3z 18)° (y 18)° 72° 3y ° x° 108, 36, 30 © Glencoe/McGraw-Hill (y 12)° x° z° 90, 93, 15 T21 Geometry: Concepts and Applications NAME ______________________________________DATE __________PERIOD______ 4-4 Study Guide Student Edition Pages 162–167 Proving Lines Parallel Suppose two lines in a plane are cut by a transversal. With enough information about the angles that are formed, you can decide whether the two lines are parallel. IF THEN corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, consecutive interior angles are supplementary, the lines are perpendicular to the same line, the lines are parallel. Example: If ⬔1 ⫽ ⬔2, which lines must be parallel? Explain. AC 储 BD because a pair of corresponding angles are congruent. Find x so that a b. 1. 2. a 110 ° a b 3. a (3x 50)° b (2x 5)° b (4x 10)° 4. 15 (4x 22)° a b 17 45 5. a 6. 57 ° 2x ° 21 a (6x 7) ° (3x 38)° b b (3x 9)° (6x 12)° 44 15 Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 7. ⬔1 ⬵ ⬔8 c d, alternate exterior angles congruent 9. m⬔7 ⫹ m⬔13 ⫽ 180 e f, consecutive interior angles supplementary © Glencoe/McGraw-Hill 8. ⬔4 ⬵ ⬔9 e f, alternate interior angles congruent 10. ⬔9 ⬵ ⬔13 c d, corresponding angles congruent T22 e f c d 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry: Concepts and Applications 4-5 NAME ______________________________________DATE __________PERIOD______ Study Guide Student Edition Pages 168–173 Slope To find the slope of a line containing two points with coordinates (x1, y1) and ( x2, y2), use the following formula. y ⫺y x2 ⫺ x1 2 1 , where x1 ⫽ x2 m⫽ The slope of a vertical line, where x1 ⫽ x2, is undefined. Two lines have the same slope if and only if they are parallel and nonvertical. Two nonvertical lines are perpendicular if and only if the product of their slopes is ⫺1. Example: Find the slope of the line ᐉ passing through A(2, ⫺5) and B(⫺1, 3). State the slope of a line parallel to ᐉ. Then state the slope of a line perpendicular to ᐉ. Let ( x1, y1) ⫽ (2, ⫺5) and ( x2, y2) ⫽ (⫺1, 3). 3 ⫺ (⫺5) 8 ⫽ ⫺ᎏᎏ. Then m ⫽ ⫺ 1⫺2 3 Any line in the coordinate plane parallel to ᐉ has slope ⫺ᎏ8ᎏ. 3 Since 8 3 ⫺ᎏᎏ ⭈ ᎏ3ᎏ ⫽ ⫺1, the slope of a line perpendicular to 8 the line ᐉ is ᎏ3ᎏ. 8 Find the slope of the line passing through the given points. 1. C(⫺2, ⫺4), D(8, 12) 8 5 2. J(⫺4, 6), K(3, ⫺10) 3. P(0, 12), R(12, 0) 16 ⴚ 7 4. S(15, ⫺15), T(⫺15, 0) 1 2 ⴚ ⴚ1 5. F (21, 12), G(⫺6, ⫺4) 6. L(7, 0), M(⫺17, 10) 16 27 5 12 ⴚ Find the slope of the line parallel to the line passing through each pair of points. Then state the slope of the line perpendicular to the line passing through each pair of points. 7. I(9, ⫺3), J(6, ⫺10) 7 3 , ⴚ 3 7 © Glencoe/McGraw-Hill 8. G(⫺8, ⫺12), H(4, ⫺1) 11 12 , ⴚ 12 11 9. M(5, ⫺2), T(9, ⫺6) ⴚ1, T23 1 Geometry: Concepts and Applications NAME ______________________________________DATE __________PERIOD______ 4-6 Study Guide Student Edition Pages 174–179 Equations of Lines You can write an equation of a line if you are given • the slope and the coordinates of a point on the line, or • the coordinates of two points on the line. Example: Write the equation in slope-intercept form of the line that has slope 5 and an x-intercept of 3. Since the slope is 5, you can substitute 5 for m in y ⫽ mx ⫹ b. y ⫽ 5x ⫹ b Since the x-intercept is 3, the point (3, 0) is on the line. y ⫽ 5x ⫹ b 0 ⫽ 5(3) ⫹ b y ⫽ 0 and x ⫽ 3 0 ⫽ 15 ⫹ b Solve for b. b ⫽ ⫺15 So the equation is y ⫽ 5x ⫺ 15. If you know two points on a line, you will need to find the slope of the line passing through the points and then write the equation. Write the equation in slope-intercept form of the line that satisfies the given conditions. 1. m ⫽ 3, y-intercept ⫽ ⫺4 y ⴝ 3x ⴚ 4 2. m ⫽ ⫺ᎏ2ᎏ, x-intercept ⫽ 6 5 2 y ⴝ ⴚ2x ⴙ 1 5 3. passes through (⫺5, 10) and (2, 4) 4. passes through (8, 6) and (⫺3, ⫺3) 0 y ⴝ ⴚ6x ⴙ 4 7 y ⴝ 9x ⴚ 6 7 5. perpendicular to the y-axis, passes through (⫺6, 4) 11 x ⴝ ⴚ7 8. perpendicular to the graph of y ⫽ 4x ⫺ 1 and passes through (6, ⫺3) y ⴝ ⴚ1x ⴚ 3 y ⴝ 3x ⴙ 18 © Glencoe/McGraw-Hill 11 6. parallel to the y-axis, passes through (⫺7, 3) yⴝ4 7. m ⫽ 3 and passes through (⫺4, 6) 5 4 T24 2 Geometry: Concepts and Applications