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## Parallels

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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**What You'll Learn 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 distance apart are ____________. parallel lines No two parallel lines intersect, no matter how far you extend them.**Parallel Lines and Planes**intersect**Q**R Parallel Lines and Planes S P K L M J 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 _____________. Plane PSR || plane JML Plane JPQ || plane MLR Plane PJM || plane QRL**Parallel Lines and Planes**Sometimes lines that do not intersect are not in the same plane. These lines are called __________. skew lines**2) All segments that intersect**B C A D 3) All segments parallel to 4) All segments skew to Parallel Lines and Planes F G DH, CG, FG, EH AB, GH, EF AD, CD, GH, AH, EH E H Name the parts of the figure: 1) All planes parallel to plane ABF Plane DCG**Parallel Lines and Transversals**What You'll Learn You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel linesand a transversal.**A**l m B is an example of a transversal. It intercepts lines l and m. Parallel Lines and Transversals In geometry, a line, line segment, or ray that intersects two or more lines at different points is called a __________ transversal 2 1 4 3 5 6 8 7 Note all of the different angles formed at the points of intersection.**Parallel Lines**Nonparallel Lines b l 2 1 2 1 3 4 4 3 m c 6 5 6 5 8 7 7 8 t r t is a transversal for l and m. r is a transversal for b and c. Parallel Lines and Transversals The lines cut by a transversal may or may not be parallel.**Exterior**Interior Exterior 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.**l**2 1 4 3 m 6 5 8 7 Parallel Lines and Transversals When a transversal intersects two lines, _____ angles are formed. eight These angles are given special names. t Exterior angles lie outside the two lines. Interior angles lie between 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**congruent 2 1 4 3 6 5 7 8**2**1 4 3 6 5 8 7 Parallel Lines and Transversals supplementary**2**1 4 3 6 5 8 7 Parallel Lines and Transversals congruent ?**End of Lesson**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**What You'll Learn You will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and a transversal.**l**2 1 4 3 m 6 5 8 7 t Transversals and Corresponding Angles When a transversal crosses two lines, the intersection creates a number ofangles 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 __________________. Give three other pairs of corresponding angles that are formed: 4 and 8 3 and 7 2 and 6**Transversals and Corresponding Angles**congruent**Transversals and Corresponding Angles**Types of angle pairs formed when a transversal cuts two parallel lines. consecutive interior alternate interior alternate exterior corresponding**s**t c 1 3 4 2 5 6 7 8 9 12 11 d 10 14 13 15 16 Transversals and Corresponding Angles s || t and c || d. Name all the angles that arecongruent to 1. Give a reason for each answer. corresponding angles 3 1 vertical angles 6 1 alternate exterior angles 8 1 corresponding angles 9 1 alternate exterior angles 14 1 corresponding angles 11 9 1 corresponding angles 16 14 1**End of Lesson**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**What You'll Learn You will learn to identify conditions that produce parallel lines. Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24). hypothesis Within those statements, we identified the “__________” and 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): IF ___________________________________, THEN ________________________________________. two parallel lines are cut by a transversal two parallel lines are cut by a transversal each pair of corresponding angles is congruent each pair of corresponding angles is congruent 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 ____________________________________.**1**a 2 b Proving Lines Parallel parallel If 1 2, then _____ a || b**a**1 2 b Proving Lines Parallel parallel If 1 2, then _____ a || b**1**a b 2 Proving Lines Parallel parallel If 1 2, then _____ a || b**a**1 2 b Proving Lines Parallel parallel If 1 + 2 = 180, then _____ a || b**t**a b Proving Lines Parallel parallel If a t andb t, then _____ a || b**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. • 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.**Y**G 90° R D 90° A Proving Lines Parallel Identify any parallel segments. Explain your reasoning.**Find the value for x so BE || TS.**E B (2x + 10)° (6x - 26)° (5x + 2)° T S ES is a transversal for BE and TS. If mBES + mEST = 180, then BE || TS by Theorem 4 – 7. Thus, if x = 24, then BE || TS. Proving Lines Parallel mBES + mEST = 180 (2x + 10) + (5x + 2) = 180 consecutive interior BES and EST are _________________ angles. 7x + 12 = 180 7x = 168 x = 24**End of Lesson**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**What You'll Learn You will learn to find the slopes of lines and use slope to identify parallel and perpendicular lines.**There has got to be some “measurable” way to get this**aircraftto clear such obstacles. Discuss how you might radio a pilot and tell him or her how toadjust the slope of their flight path in order to clear the mountain. If the pilot doesn’t change something, he / she will not make it home for Christmas. Would you agree? Consider the options: 1) Keep the same slope of his / her path. Not a good choice! 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” requiredand divide that by the “horizontal change” required.**y**10000 x 0 0 10000**y**10 10 5 x -10 -10 -5 5 10 10 -5 -10 -10 Slope The steepness of a line is called the _____. slope Slope is defined as the ratio of the ____, or vertical change, to the ___, orhorizontal change, as you move from one point on the line to another. run rise**y**x Slope The slope m of the non-vertical line passing through the pointsand is**y**(3, 6) (1, 1) ? 6 & 7 x Slope The slopem of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right. rise = 6 - 1 = 5 units run = 3 - 1 = 2 units**Slope**the same slope**?**8 & 9 Slope the product of their slope is -1**End of Lesson**Practice Problems: 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 20, 22, 24, 26, 30, and 32 (total = 19)**y**8 8 7 6 5 4 (3, 5) (2, 3) 3 (1, 1) 2 1 x 0 -1 -1 -1 -1 1 2 3 4 5 6 7 8 8 0 Equations of Lines What You'll Learn You will learn to write and graph equations of lines. linear equation The equation y = 2x – 1 is called a _____________ because its graph is a straight line. We can substitute different values for x in the graph to find corresponding values for y. There are many more points whose ordered pairs are solutions of y = 2x – 1. These points also lie on the line. 1 1 y = 2(1) -1 2 3 y = 2(2) -1 3 5 y = 2(3) -1**y = 2x – 1**y 5 5 y - intercept slope 4 (0, -1) 3 2 1 x 0 -1 -2 -3 -3 -3 -3 -2 -1 1 2 3 4 5 5 0 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 This value is called the ____________ of the line. y = mx + b Most linear equations can be written in the form __________. slope – intercept form This form is called the ___________________. y = mx + b**y**5 5 4 (0, 3) (1, 1) 3 2 1 x 0 -1 -2 -3 -3 -3 -3 -2 -1 1 2 3 4 5 5 0 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 1) Identify and graph the y-intercept. 2) Follow the slope a second point on the line. 3) Draw the line between the two points.**3) Write an equation of the line perpendicualr to the graph**of that passes through the point ( - 3, 8). 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 y = -4x -4**End of Lesson**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)