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 lines and a transversal.

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 B A l m 1 2 4 3 5 6 8 7 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. l m 1 2 3 4 5 7 6 8 Parallel Lines t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c Nonparallel Lines r is a transversal for b and c. r

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

Parallel Lines and Transversals When a transversal intersects two lines, _____ angles are formed. eight These angles are given special names. l m 1 2 3 4 5 7 6 8 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 Alternate Interior Angles If two parallel lines are cut by a transversal, then each pair of Alternate interior angles is _________. congruent 1 2 4 3 5 6 8 7

Parallel Lines and Transversals Theorem 4-2 Consecutive Interior Angles If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is _____________. supplementary 1 2 3 4 5 7 6 8

Parallel Lines and Transversals Theorem 4-3 Alternate Exterior Angles If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is _________. congruent 1 2 3 4 5 7 6 8 ?

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. Angle 1 and 5 are called __________________. corresponding angles l m 1 2 3 4 5 7 6 8 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 Corresponding Angles If two parallel lines are cut by a transversal, then each pair of corresponding angles is _________. congruent

End of Lesson

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.

Transversals and Corresponding Angles Concept Summary Congruent Supplementary Types of angle pairs formed when a transversal cuts two parallel lines. alternate interior consecutive interior alternate exterior corresponding

Transversals and Corresponding Angles 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 s || t and c || d. Name all the angles that are congruent to 1. Give a reason for each answer. 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

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). Within those statements, we identified the “__________” and the “_________”. hypothesis 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 ____________________________________.

Proving Lines Parallel Postulate 4-2 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 1 2 a b If 1 2, then _____ a || b

Proving Lines Parallel Theorem 4-5 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 are _______. parallel 1 2 a b If 1 2, then _____ a || b

Proving Lines Parallel Theorem 4-6 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 are _______. parallel 1 2 a b If 1 2, then _____ a || b

Proving Lines Parallel Theorem 4-7 In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two lines are _______. parallel 1 2 a b If 1 + 2 = 180, then _____ a || b

Proving Lines Parallel Theorem 4-8 In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two lines are _______. parallel If a  t and b  t, then _____ a b t a || b

Proving Lines Parallel We now have five ways to prove that two lines are parallel. Concept Summary 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.

Proving Lines Parallel Identify any parallel segments. Explain your reasoning. G A Y D R 90°

Proving Lines Parallel Find the value for x so BE || TS. E B S T (6x - 26)° (2x + 10)° (5x + 2)° ES is a transversal for BE and TS. mBES + mEST = 180 BES and EST are _________________ angles. (2x + 10) + (5x + 2) = 180 consecutive interior 7x + 12 = 180 If mBES + mEST = 180, then BE || TS by Theorem 4 – 7. 7x = 168 x = 24 Thus, if x = 24, then BE || TS.

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 aircraft to clear such obstacles. Discuss how you might radio a pilot and tell him or her how to adjust 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” required and divide that by the “horizontal change” required.

y x 10000

The steepness of a line is called the _____. slope Slope is defined as the ratio of the ____, or vertical change, to the ___, or horizontal change, as you move from one point on the line to another. rise run y x 10 -5 -10 5

The slope m of the non-vertical line passing through the points and is y x

The slope “m” of a line containing two points with coordinates Definition of Slope The slope “m” of a line containing two points with coordinates (x1, y1), and (x2, y2), is given by the formula

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 x (3, 6) rise = 6 - 1 = 5 units (1, 1) run = 3 - 1 = 2 units ? 6 & 7

Slope Postulate 4 – 3 Two distinct nonvertical lines are parallel iff they have _____________. the same slope

Slope Postulate 4 – 4 Two nonvertical lines are perpendicular iff ___________________________. the product of their slope is -1 ? 8 & 9

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)

Equations of Lines What You'll Learn You will learn to write and graph equations of lines. The equation y = 2x – 1 is called a _____________ because its graph is a straight line. linear equation We can substitute different values for x in the graph to find corresponding values for y. y x 8 1 3 5 7 -1 2 4 6 x y = 2x -1 y There are many more points whose ordered pairs are solutions of y = 2x – 1. These points also lie on the line. (3, 5) 1 y = 2(1) -1 1 2 (2, 3) y = 2(2) -1 3 3 y = 2(3) -1 5 (1, 1)

y = mx + b Look at the graph of y = 2x – 1 . Equations of Lines Look at the graph of y = 2x – 1 . The y – value of the point where the line crosses the y-axis is ___. - 1 This value is called the ____________ of the line. y - intercept Most linear equations can be written in the form __________. y = mx + b This form is called the ___________________. slope – intercept form y x 5 -2 1 3 -3 2 -1 4 y = 2x – 1 y = mx + b slope y - intercept (0, -1)

An equation of the line having slope m and y-intercept b is y = mx + b Equations of Lines Slope – Intercept Form An equation of the line having slope m and y-intercept b is y = mx + b

1) Rewrite the equation in slope – intercept form by solving for y. 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 x 5 -2 1 3 -3 2 -1 4 1) Identify and graph the y-intercept. (0, 3) 2) Follow the slope a second point on the line. (1, 1) 3) Draw the line between the two points.

3) Write an equation of the line perpendicualr to the graph of 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

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)