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Parallels § 4.1 Parallel Lines and Planes § 4.2 Parallel Lines and Transversals § 4.3 Transversals and Corresponding Angles § 4.4 Proving Lines Parallel § 4.5 Slope § 4.6 Equations of Lines Parallel Lines and Planes You will learn to describe relationships among lines, parts of lines, and planes. In geometry, two lines in a plane that are always the same parallel lines distance apart are ____________. No two parallel lines intersect, no matter how far you extend them. Parallel Lines and Planes Definition of Parallel Lines Two lines are parallel iff they are in the same plane and intersect do not ________. Parallel Lines and Planes Planes can also be parallel. The shelves of a bookcase are examples of parts of planes. The shelves are the same distance apart at all points, and do not appear to intersect. parallel They are _______. parallel planes In geometry, planes that do not intersect are called _____________. Q R Plane PSR || plane JML Plane JPQ || plane MLR Plane PJM || plane QRL P J K S L M Parallel Lines and Planes Sometimes lines that do not intersect are not in the same plane. skew lines These lines are called __________. Definition of Skew Lines Two lines that are not in the same plane are skew iff they do not intersect. Parallel Lines and Planes Name the parts of the figure: 1) All planes parallel to plane ABF Plane DCG B 2) All segments that intersect DH AD, CD, GH, AH, EH C A D 3) All segments parallel to CD AB, GH, EF 4) All segments skew to AB DH, CG, FG, EH F E G H Parallel Lines and Transversals You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal. Parallel Lines and Transversals In geometry, a line, line segment, or ray that intersects two or more lines at transversal different points is called a __________ A 2 1 4 5 8 6 7 3 l m B AB is an example of a transversal. It intercepts lines l and m. Note all of the different angles formed at the points of intersection. Parallel Lines and Transversals Definition of Transversal In a plane, a line is a transversal iff it intersects two or more Lines, each at a different point. The lines cut by a transversal may or may not be parallel. Parallel Lines Nonparallel Lines l 1 2 4 3 lm t 1 2 4 3 m 5 6 8 7 c 5 6 8 7 b || c t is a transversal for l and m. b r r is a transversal for b and c. Parallel Lines and Transversals Two lines divide the plane into three regions. The region between the lines is referred to as the interior. The two regions not between the lines is referred to as the exterior. Exterior Interior Exterior Parallel Lines and Transversals eight angles are formed. When a transversal intersects two lines, _____ These angles are given special names. l 1 2 4 3 m 5 6 8 7 t Interior angles lie between the two lines. Exterior angles lie outside the two lines. Alternate Interior angles are on the opposite sides of the transversal. Alternate Exterior angles are on the opposite sides of the transversal. Consectutive Interior angles are on the same side of the transversal. Parallel Lines and Transversals Theorem 4-1 If two parallel lines are cut by a transversal, then each pair of congruent Alternate Alternate interior angles is _________. Interior Angles 1 2 4 3 5 6 8 7 4 6 3 5 Parallel Lines and Transversals Theorem 4-2 If two parallel lines are cut by a transversal, then each pair of supplementary Consecutive consecutive interior angles is _____________. Interior Angles 1 2 4 3 5 6 8 7 4 5 180 3 6 180 Parallel Lines and Transversals Theorem 4-3 If two parallel lines are cut by a transversal, then each pair of congruent Alternate alternate exterior angles is _________. Exterior Angles 1 2 4 3 5 6 8 7 1 7 2 8 Practice Problems: 1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, and 46 (total = 23) Transversals and Corresponding Angles You will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and a transversal. Transversals and Corresponding Angles When a transversal crosses two lines, the intersection creates a number of angles that are related to each other. Note 1 and 5 below. Although one is an exterior angle and the other is an interior angle, both lie on the same side of the transversal. corresponding angles Angle 1 and 5 are called __________________. l 1 2 4 3 m 5 6 8 7 t Give three other pairs of corresponding angles that are formed: 4 and 8 3 and 7 2 and 6 Transversals and Corresponding Angles Postulate 4-1 If two parallel lines are cut by a transversal, then each pair of congruent Corresponding corresponding angles is _________. Angles Transversals and Corresponding Angles Types of angle pairs formed when a transversal cuts two parallel lines. Concept Summary Congruent Supplementary alternate interior consecutive interior alternate exterior corresponding Transversals and Corresponding Angles s s || t and c || d. Name all the angles that are congruent to 1. Give a reason for each answer. 1 2 5 6 9 10 13 14 3 1 corresponding angles 6 1 vertical angles 8 1 alternate exterior angles 9 1 corresponding angles 14 1 alternate exterior angles 11 9 1 corresponding angles 16 14 1 corresponding angles t 3 7 11 12 15 16 c 4 8 d Practice Problems: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, and 38 (total = 19) Proving Lines Parallel You will learn to identify conditions that produce parallel lines. Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24). hypothesis and the Within those statements, we identified the “__________” conclusion “_________”. I said then that in mathematics, we only use the term “if and only if” if the converse of the statement is true. Proving Lines Parallel Postulate 4 – 1 (pg. 156): two parallel lines are cut by a transversal IF ___________________________________, each pair of corresponding angles is congruent THEN ________________________________________. The postulates used in §4 - 4 are the converse of postulates that you already know. COOL, HUH? §4 – 4, Postulate 4 – 2 (pg. 162): IF ________________________________________, THEN ____________________________________. Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel _______. Postulate 4-2 1 2 If 1 2, a b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines parallel are _______. Theorem 4-5 If 1 2, a 2 1 b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines parallel are _______. Theorem 4-6 1 2 If 1 2, a b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two parallel lines are _______. Theorem 4-7 If 1 + 2 = 180, 1 2 a b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two parallel lines are _______. Theorem 4-8 If a t and b t, t a b a || b then _____ Proving Lines Parallel We now have five ways to prove that two lines are parallel. Show that a pair of corresponding angles is congruent. Show that a pair of alternate interior angles is congruent. Concept Summary Show that a pair of alternate exterior angles is congruent. Show that a pair of consecutive interior angles is supplementary. Show that two lines in a plane are perpendicular to a third line. Proving Lines Parallel Identify any parallel segments. Explain your reasoning. GY and RD are both perpendicular toGA therefore, GY RD by T heorem4 - 8. G R A Y 90° 90° D Proving Lines Parallel B Find the value for x so BE || TS. T (6x - 26)° (2x + 10)° (5x + 2)° ES is a transversal for BE= and mBES + mEST 180TS. (2x + and 10) EST + (5x are + 2)_________________ = 180 consecutive interior angles. BES 7x + 12 = 180 If mBES + mEST = 180, then BE || TS by Theorem 4 – 7. 7x = 168 x = 24 Thus, if x = 24, then BE || TS. S E Practice Problems: 1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 25, and 26 (total = 19) Slope You will learn to find the slopes of lines and use slope to identify parallel and perpendicular lines. There has the got to bechange some “measurable” to will get not this make aircraft If the pilot doesn’t something, heway / she it Consider options: to clearfor such obstacles. home Christmas. Would you agree? 1) Keep the same slope of his / her path. Discuss you might radio a pilot and tell him or her how to Not ahow good choice! adjust the slope of their flight path in order to clear the mountain. 2) Go straight up. Not possible! This is an airplane, not a helicopter. Fortunately, there is a way to measure a proper “slope” to clear the obstacle. We measure the “change in height” required and divide that by the “horizontal change” required. vertical change y Slope horizontal change x x y vertical change y 4, 000 ft 4 2 Slope horizontal change x 10, 000 ft 10 5 y 10000 0 0 10000 x Slope slope The steepness of a line is called the _____. Slope is defined as the ratio of the ____, run or rise or vertical change, to the ___, horizontal change, as you move from one point on the line to another. y 1 0 5 -10 -5 5 -5 -10 1 0 x Slope The slope m of the non-vertical line passing through the points ( x1 , y1 ) and ( x2 , y2 ) is y ( x2 , y2 ) y2 y1 ( x1 , y1 ) x2 x1 x m rise change in y y2 y1 run change in x x2 x1 Slope The slope “m” of a line containing two points with coordinates (x1, y1), and (x2, y2), is given by the formula Definition of Slope differenceof they - coordinates slope differenceof thecorresponding x - coordinates y2 y1 m , x 2 x1 where x2 x1 Slope The slope m of a non-vertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. y rise (y ) m run (x) (3, 6) 6 1 m 3 1 5 m 2 rise = 6 - 1 = 5 units (1, 1) x run = 3 - 1 = 2 units 6 & 7 Slope Two distinct nonvertical lines are parallel iff they have the same slope _____________. Postulate 4–3 y1 m1 x b1 y2 m2 x b2 L1 L2 iff m1 m2 Slope Two nonvertical lines are perpendicular iff the product of their slope is -1 ___________________________. Postulate 4–4 y1 m1 x b1 L1 L2 iff y2 m2 x b2 m1 m2 1 8 & 9 Practice Problems: 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 20, 22, 24, 26, 30, and 32 (total = 19) Equations of Lines You will learn to write and graph equations of lines. linear equation because its graph is The equation y = 2x – 1 is called a _____________ a straight line. We can substitute different values for x in the graph to find corresponding values for y. y 8 7 6 x y = 2x -1 y There are many more points whose ordered 1 pairs1are solutions y = 2(1) of -1 y = 2x – 1. These points also lie on the line. 2 3 y = 2(2) -1 3 y = 2(3) -1 5 5 (3, 5) 4 3 (2, 3) 2 1 (1, 1) x 0 -1 -1 0 1 2 3 4 5 6 7 8 Equations of Lines Look at the graph of y = 2x – 1 . -1 The y – value of the point where the line crosses the y-axis is ___. y - intercept of the line. This value is called the ____________ y = mx + b Most linear equations can be written in the form __________. – intercept form This form is called the slope ___________________. y = mx + b slope y = 2x – 1 y 5 4 3 y - intercept 2 1 x 0 -1 (0, -1) -2 -3 -3 -2 -1 0 1 2 3 4 5 Equations of Lines Slope – Intercept Form An equation of the line having slope m and y-intercept b is y = mx + b Equations of Lines 1) Rewrite the equation in slope – intercept form by solving for y. 2x – 3 y = 18 2) Graph 2x + y = 3 using the slope and y – intercept. y = –2x + 3 y 5 4 1) Identify and graph the y-intercept. 3 2) Follow the slope a second point on the line. 3) Draw the line between the two points. (0, 3) 2 1 (1, 1) x 0 -1 -2 -3 -3 -2 -1 0 1 2 3 4 5 Equations of Lines 1) Write an equation of the line parallel to the graph of y = 2x – 5 that passes through the point (3, 7). y = 2x + 1 2) Write an equation of the line parallel to the graph of 3x + y = 6 that passes through the point (1, 4). y = -3x + 7 3) Write an equation of the line perpendicualr to the graph of that passes through the point ( - 3, 8). y = -4x -4 1 y x5 4 Practice Problems: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 40, and 42 (total = 24)