Geometry. Unit 3. relationships and slope. Essential Questions. o When does algebra help me understand geometry, and when does

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1 Geometry Unit 3 Parallel and P rpendicular Lines This unit focuses on exploring the relationships of angles formed by lines cut by a transversal (including parallel lines), and writing equations of lines which include parallel and perpendicular lines. Students should use multiple approaches to draw valid conclusions including angle pair relationships and slope. Essential Questions o When does algebra help me understand geometry, and when does geometry help me understand algebra? How do I "show what I know" mathematically? Why do I show it this way? How can concrete models, patterns, and relationships help me describe and explain concepts in geometry?

2 Unit 3 Parallel Lines and Transversals(3.1) lines - two lines that never have the same. and will We use on the lines to indicate they are. (11) D BC//AD Two planes can also or be lines - two lines are are if they intersect and E!: Using the figure at the right: a. Name ell segments that are skew to 6ÿ, b. Name all planes parallel to the plane ADH, A B c. Name all segments that are parallel to AT. d. Name all segments that intersect T K H

3 the given line., then there is : If there is a and a one line through the point not on the to If,then There is exactly one line through P parallel to line e the givenline., then there is :If there is a and a one line through the point not on the to If,then There is exactly one line through P perpendicular to line - a line that intersects two or more coplanar lines at points.

4 When two or more lines are cut by a transversal they form special angles: : have corresponding positions. Sit in the savne spot on the other line. short hand: l) : lie between the two lines and on the of the transversal. short hand: : lie outside the two lines on of the transversal. sides short hand: : lie between the two lines on sides of the transversal. short hand:

5 EX 1: Look at the figure below. Transversa! t intersects lines m and n Consecutive Interior Angles: 3 5 Alternate Interior Angles: Alternate Exterior Angles: Corresponding Angles: EX 2: Use the figure at the flight to answer the following: laÿ Name the transversal to lines n & p b. Name the transversal to lines m & p c. Name the transversal to lines m & n d. Identify each pair of langfes: <7 & <12 <8 & <10 <2 & <12 <1&<11 3 m <6 & <5 <6 & <11

6 Unit 3 Use Parallel Lines and Transversals (3.2) In 3,1 you identified the special types of angles that are formed when two or more lines are cut by a transversal. Let's review them: Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Today we are going to talk about what happens when two lines are cut by a. When this happens we can identify their and they are If two parallel lines are cut by a transversal, then the following will happen: Correspondin9 Angles Postulate If two angles are lines are cut by a l, then each pair of The postulate can be written as: the lines are parallel," <4 _ÿ<6 Alternate Interior Angles Theorem If two angles are lines are cut by a then each pair of The theorem can be written as: the lines are parallel.'. <4 -ÿ<6

7 Alternate Exterior An qles Theorem If two angles are lines are cut by a.,, then each pair of The theorem can be written as: the lines are paralle!,', <4 ÿ<6 Consecutive Interior Anqles Theorem If two angles are lines are cut by a, then each pair of The theorem can be written as: \ the lines are parallel Y7 ÿ ÿ--f----7-ÿ,', rn<4 + rn<6 : 180o EX 1: In the figure, //rn and c//d, Find the values of x, y, and Z, X -- y= m xj >d 3y ÿ 8

8 EX 2: In the picture below m//n and pi/q. Find the measure of each angle if m<2 = 350 P hi<l: m<9: 5 6 m<4= m<li= 9 ÿ rfl <5 =.m <12 = m <6 = ÿm <13 = m <7 = m <14 = I11 <8 = m <15 = m<16: EX 3: In the picture bejow m//n. Find the measure of each angle if m<20 : 650 and m<14 = 700 P q m<lo= X 3 4 re<l= m<2:ÿ m<3: m <11 : m <12 = 15 m <119 = Given: p II q Prove: <1 ÿ <2 Prove the Alternate Interior Angles Theorem /t ' y X pp Statements easons

10 Unit 3 Prove Lines are Parallel(3.3) You have Jearned so far that if two or more lines are parallel and are cut by a transversal, thee the following are true: Corresponding angles are congruent Remember: //Iines ÿ corrÿ <'s Alternate interior angles are congruent Remember: //Fines ==ÿ> AIA's Alternate exterior angles are congruent Remember: //lines =zÿ> AEA's Consecutive interior angles are supplementary Remember: //lines zÿ> CIA's suppl, Today you will learn that the also! of the above postulate and theorems are If Corresponding Angles Converse If two lines are cut by a so the l, then the are angles are The postulate can be written as: <4 ÿ- <6ÿ \.'. the lines are parallel Alternate Interior Angles Converse If two lines are cut by a so the, then the are angles are The theorem can be written as: f ÿ <4 ÿ <6 / 6ÿ.'. the lines are parallel ÿ/ÿ7- / / J

11 Alternate Exterior Angles Converse If two lines are cut by a so the, then the are angles are The theorem can be written as:,'. the lines are parallel \ J Consecutive Interior Angles Converse If two lines are cut by a so the, then the are angles are The theorem can be written as: Let 1ÿ<4 = 40ÿ and mÿ6 = 140 \ m<4 + m<6 = // 1800 ÿ ÿ,'.the lines are parallel EX 1: Can you conclude m//n? Explain why or why not?

12 EX 2: Given: <1 = <2 Prove: p//q >P q Statements Reasons,, 2. <2 ÿ < <1 ÿ <3 3, Paraqraph Proof A proof can also be written in paragraph form, called a statements and reasons in a paragraph proof are written in sentences, using words to explain the logical flow of the argument. The EX3: In the figure, r//s and <1ÿ <3. Prove: p//q q Look at the diagram to make a plan. The diagram suggests that you look at angles 1, 2, and 3. Also, you may find it helpful to focus on one pair of lines and one transversal at a time. a. Look at <1 and <2 b. Look at <2 and <3 r S r S _ I I <1ÿ <2 because r//s If <2 m <3, then p//q

14 Unit 3 Find and Use Slopes of Lines(3.4) Remember: You can find of a line by: 1. Counting the.(1 ) over the.) on the graph: 2. Use the slope formula: EX 1: Given the following coordinates, use the slope formula to calculate the slope of line AB. a) A(4, 8) B(7,-2) b) A(8, 8) B(8, 5) c) A(-2, 2) B(2, 2) Slopes of Parallel Lines Postulate In a coordinate plane, two nonvertical lines are b) What does this mean? c) if and only if they have the Any hvo vertical lines haw the ÿame slope. They wÿll have what kind of slope? slope of line a = slope of line b Slopes of Perpendicular Lines Postulate In a coordinate plane, two nonvertical lines are (multiply) of their is -1 if and only if the /jÿ'-. / **If the product of ÿo [ numbers is oi, ÿhen ÿhe numbers ore called ÿ ) Horlzontol lines ore perpemdicu/or to vertical llaes, What will be their slopes? slope of line a o slope of line b = -1

16 Unit 3 Write and Graph Equations of Lines(3,5) Linear equations may be written in different forms. Slope Intercept Form: Yÿ where m = and b = written as an. The y-int, is always where x is always. Ex: (0o b) EX 1: What is the slope and y-intercept of y = ÿx - 6 m = y-int: Point-Slope Form: where (xl,yl) is a point on the line and m = Standard Form: where A, B, and C are real numbers and A and B are not both zero. A is also positive.

17 EX 2: Write an equation in slope intercept form of the line passing through the point (-1, 1) that is parallel to the line with the equation y = 2x - 3. Graph both lines. 'i! ÿ i 1ÿ ' ' i ÿ i iii ii: i: il! ilÿ il i Iÿ ÿ,ÿii,i l!:i!ÿ i! i : i ii ii ÿ ÿ ii i! i! i' ' ij iÿ i ill i! i Y i'ÿ! il I: Iiÿ I ii ÿiÿ ii I I!: iÿ iÿ! Ii i! :i 'i ii i;: ilÿ i i,i i! il: J:: ili ÿ:i II i:l I i!,! 1!i 111! i! jl: J i1 :il EX 3: Write an equation in point-slope form of the line passing through the point (2, 3) that is perpendicular to the line with the equation 2x + 4y = 6, Graph the lines. i Jl j:i: i:ll [ i! i'il i I: I/ if! iÿ! i'ÿ L ij li' IIÿ l! l! :I i:! I1 i1!,l EX 4: Write an equation of a line in standard form that passes through the points (2, 2) and (5, 3)

18 You can graph a line from standard form by finding the x and y intercepts. E 5: Graph 3x 4y - 12 using x and y intercepts,!,i 1i [,:!ÿ ] i ii ' ], I iij i] I II :i I:! Ill i

20 Unit 3 Prove Theorems About Perpendicular Lines(3.6) Theorem: If two lines intersect to form a are of congruent angles, then the lines What does this mean about <1 and <2? If m<l = rn<2, then.llines. Theorem: If two lines are, then they intersect to form four 8 angles. b if a_lb, then <1, <2, <3, & <4 are right angles are Theorem:,. If two sides of two acute angles are, then the angles B C If AB l BC, then <ABC is a right angle, <1 & <2 are complementary. t

21 Perpendicular Transversal Theorem: If a transversal is to the other. to one of two lines, then it is zf ),then Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are to each other. to the line, then they are parallel If <_.--- ::3 ÿ, then Distance from a line: The distance from a segment to a NOT on the line is the to the line from that point. of the Point not ÿ Iÿ.ÿÿ. distonce on the line The between two segment joining the two lines. lines is the length of any EX 1: Use the distance formula to find the distance between two parallel lines.!i Il li Hint: Find the slope.

Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines

Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines Two lines are if they are coplanar and do not intersect. Skew lines. Two