 § 4.1 Parallel Lines and Planes Parallel Lines and PlanesParallel Lines and Planes  § 4.4 Proving Lines Parallel Proving Lines ParallelProving Lines Parallel  § 4.3 Transversals and Corresponding Angles Transversals and Corresponding AnglesTransversals and Corresponding Angles  § 4.2 Parallel Lines and Transversals Parallel Lines and TransversalsParallel Lines and Transversals  § 4.6 Equations of Lines Equations of LinesEquations of Lines  § 4.5 Slope Slope

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.

Definition of Parallel Lines Two lines are parallel iff they are in the same plane and do not ________. intersect

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. They are _______. parallel In geometry, planes that do not intersect are called _____________. parallel planes Q J K M L S R P Plane PSR || plane JML Plane JPQ || plane MLR Plane PJM || plane QRL

Sometimes lines that do not intersect are not in the same plane. These lines are called __________. skew lines Definition of Skew Lines Two lines that are not in the same plane are skew iff they do not intersect.

A C B E G H D F Name the parts of the figure: 1) All planes parallel to plane ABF 2) All segments that intersect 3) All segments parallel to 4) All segments skew to Plane DCG AD, CD, GH, AH, EH AB, GH, EF DH, CG, FG, EH

You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal.

In geometry, a line, line segment, or ray that intersects two or more lines at different points is called a __________ transversal l m B A is an example of a transversal. It intercepts lines l and m. Note all of the different angles formed at the points of intersection

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 Parallel Lines t is a transversal for l and m. t b c Nonparallel Lines r is a transversal for b and c. r

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

l m When a transversal intersects two lines, _____ angles are formed. eight These angles are given special names. 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. Consectutive Interior angles are on the same side of the transversal. Alternate Exterior angles are on the opposite sides of the transversal.

Theorem 4-1 Alternate Interior Angles If two parallel lines are cut by a transversal, then each pair of Alternate interior angles is _________ congruent

Theorem 4-2 Consecutive Interior Angles If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is _____________. supplementary

Theorem 4-3 Alternate Exterior Angles If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is _________. congruent

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)

You will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and a transversal.

l m t 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 Give three other pairs of corresponding angles that are formed:  4 and  8  3 and  7  2 and  6

Postulate 4-1 Corresponding Angles If two parallel lines are cut by a transversal, then each pair of corresponding angles is _________. congruent

Concept Summary CongruentSupplementary alternate interior alternate exterior corresponding consecutive interior Types of angle pairs formed when a transversal cuts two parallel lines.

s t c d 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  11   9   1 corresponding angles  14   1 alternate exterior angles  16   14   1 corresponding angles

Practice Problems: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, and 38 (total = 19)

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.

Postulate 4 – 1 (pg. 156): IF ___________________________________, THEN ________________________________________. two parallel lines are cut by a transversal 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 ____________________________________. each pair of corresponding angles is congruent two parallel lines are cut by a transversal

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 If  1  2, then _____ a || b 1 2 a b

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 If  1  2, then _____ a || b 1 2 a b

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 If  1  2, then _____ a || b 1 2 a b

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 If  1 +  2 = 180, then _____ a || b 1 2 a b

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 a b t

Concept Summary We now have five ways to prove that two lines are parallel. SShow that a pair of corresponding angles is congruent. SShow that a pair of alternate interior angles is congruent. SShow that a pair of alternate exterior angles is congruent. SShow that a pair of consecutive interior angles is supplementary. SShow that two lines in a plane are perpendicular to a third line.

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

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

Practice Problems: 1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 25, and 26 (total = 19)

You will learn to find the slopes of lines and use slope to identify parallel and perpendicular lines.

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. 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.

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

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

Definition of Slope The slope “m” of a line containing two points with coordinates (x 1, y 1 ), and (x 2, y 2 ), is given by the formula

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

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

Postulate 4 – 4 Two nonvertical lines are perpendicular iff ___________________________. the product of their slope is -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)

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. 0 y 0 x xy = 2x -1y y = 2(1) y = 2(2) -1 y = 2(3) -1 (1, 1) (2, 3) (3, 5) There are many more points whose ordered pairs are solutions of y = 2x – 1. These points also lie on the line.

0 y 0 x y = 2x – 1 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 (0, -1) Most linear equations can be written in the form __________. y = mx + b This form is called the ___________________. slope – intercept form y = mx + b slope y - intercept

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. 2x – 3 y = 18 2) Graph 2x + y = 3 using the slope and y – intercept. y = –2x y 0 x ) Identify and graph the y-intercept. 2) Follow the slope a second point on the line. (0, 3) (1, 1) 3) Draw the line between the two points.

1) Write an equation of the line parallel to the graph of y = 2x – 5 that passes through the point (3, 7). 2) Write an equation of the line parallel to the graph of 3x + y = 6 that passes through the point (1, 4). 3) Write an equation of the line perpendicualr to the graph of that passes through the point ( - 3, 8). y = 2x + 1 y = -3x + 7 y = -4x -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)