Lines and Angles Notes Geometry Unit 3: Lesson 1. Parallel lines. Skew lines. Parallel planes. Transversal. Alternate Interior Angles t

Transcription

1 Lines and Angles Noes Geoery Uni 3: Lesson 1 Nae Parallel lines D C Skew lines A B Parallel planes E H F G Transversal Alernae Inerior Angles, l Alernae Exerior Angles, Corresponding Angles,,, Sae Side Inerior Angles (Consecuive Inerior Angles),

2 Properies of Parallel Lines Nae Geoery Uni 3: Lesson 2-1 EQ: Wha are he ypes of angles ha are fored by parallel lines? Types of Angles Fored By Parallel Lines: 1) If wo lines are parallel hen corresponding angles are congruen l 2) If wo lines are parallel hen alernae inerior angles are congruen l 3) If wo lines are parallel hen alernae exerior angles are congruen l

3 4) If wo lines are parallel hen sae side inerior angles are suppleenary l 1. If l, and 1 = 110, find he reainder of he angle easures l For probles 2-5, solve for x given ha l : 2. 5x - 20 l 3x + 10

4 3. 4x + 20 l 2x x + 20 l 4x x + 20 l 4x x + 70 l 8x + 16

5 Parallel Lines Cu by a Transversal Proofs 1 Nae Geoery Uni 3: Lesson Given: l Prove: < 1 < 2 1 l 2 2. Given: l Prove: < 1 < 2 1 l 2 3. Given: l Prove: < 1 < l 2

6 4. Given: a b Prove: 1 and 8 are suppleenary 1 a b Saeens Reasons 1. a b = and 8 are suppleenary = = and 8 are suppleenary 7.

7 Proving Lines Parallel 1 Nae Geoery Uni 3: Lesson 2-3 EQ: How do you prove ha wo lines are parallel? Two lines can be proven o be parallel when: 1) If wo lines are cu by a ransversal such ha corresponding angles are congruen, hen he lines are parallel. 2) If wo lines are cu by a ransversal such ha alernae inerior angles are congruen, hen he lines are parallel. 3) If wo lines are cu by a ransversal such ha sae side inerior angles are suppleenary, hen he lines are parallel. Exaple: Given: 1 2 Prove: l 1 l 2 Saeen Reason 1. Given: 1 3 Prove: l 1 l 3

8 2. Given: s 6 9 Prove: l s l Given: s 6 11 Prove: l s l

9 Proving Lines Parallel 2 Nae Geoery Uni 3: Lesson 2-4 EQ: How do you prove ha wo lines are parallel? Exaple: Given: l 6 11 Prove: s s l Saeen Reason 1. Given: l 1 13 Prove: s s l

10 2. Given: s 5 9 Prove: l s l Given: l 3 14 Prove: s s l

11 Proving Lines Parallel 3 Nae Geoery EQ: How do you prove ha wo lines are parallel? 1. Given: 1 and 3 are suppleenary Prove: a b a b Proof: Saeens Reasons 1. 1 and 3 are suppleenary and 2 are suppleenary a b 4.

12 2. Given: 1 and 2 are suppleenary Prove: a b Proof: Saeens Reasons Given 2 a 3 b If wo angles are suppleenary, hen hey add o 180. (1) = Subsiuion Propery (2, 4) If wo angles add o 180, hen hey are suppleenary. (5) 7. a b 7.

13 Parallel and Perpendicular Lines Geoery Uni 3: Lesson 3 Nae 1. Solve for he variables, given ha l. Wha do you know ha allowed you o solve he proble? 1. 8y l 2x x + 15 y l 4x + 20

14 3. 2y 40 7x + 18 l 3x x + 24 l y 4x

15 Angles of Triangles Geoery Uni 3: Lesson 4 Nae Through a poin no on a line, here is one and only one line parallel o he given line. P Given ABC, consruc a line hrough poin B, parallel o AC. P B R A C This auxiliary line (a line added o a diagra o help explain relaionships in proofs) can be used o explore he su of he inerior angles in a riangle. Triangle Angle-Su Theore: The su of he easures of he angles of a riangle is = Proof of Triangle Angle-Su Theore Given: ABC, PR AC Prove: A C = 180 P B R Saeens 1. ABC, PR AC Reasons 1. Given 2. PBC and 3 are suppleenary 2. Angles ha for a linear pair are suppleenary. 3. PBC + 3 = Suppleenary angles add o 180. (2) 4. PBC = Angle Addiion Posulae = Subsiuion Propery (3, 4) 6. 1 A, 3 C 6. If wo parallel lines are cu by a ransversal, hen alernae inerior angles are congruen. (1) 7. 1 = A, 3 = C 7. Congruen angles have = easure. (6) 8. A C = Subsiuion Propery (5, 7) A C

16 Exerior Angle of a Triangle: An exerior angle of a riangle is fored by exending a side of he riangle. exerior angle For each exerior angle, he wo nonadjacen inerior angles are called is reoe inerior angles. reoe inerior angle reoe inerior angle exerior angle Exerior Angle Theore: The easure of each exerior angle of a riangle equals he su of is wo reoe inerior angles. reoe inerior angle 1 4 = reoe inerior angle exerior angle Proof of Exerior Angle Theore Given: 4 is an exerior angle of he riangle Prove: 4 = Saeens Reasons 1. 4 is an exerior angle of he riangle 1. Given 2. 3 and 4 are suppleenary 2. Angles ha for a linear pair are suppleenary = Suppleenary angles add o 180. (2) = The su of he inerior angles of a riangle is = Subsiuion Propery (3, 4) 6. 3 = 3 6. Reflexive Propery 7. 4 = Subracion Propery (5, 7)

17 1. Wha are he values of x, y, and z? 2. Find he easure of each angle. 50 o 2x + 15 x Wha is he easure of 1? 4. Wha is he easure of 2?

18 5. Two angles of a riangle easure 53. Wha is he easure of an exerior angle a each verex of he riangle? 6. When radar racks an objec, he reflecion of signals off he ground can resul in cluer. Cluer causes he receiver o confuse he real objec wih is reflecion, called a ghos. In he diagra, here is a radar receiver a A, an airplane a B, and he airplane s ghos a D. Wha is he value of x?

19 Consruc Parallel and Perpendicular Lines Geoery Uni 4: Lesson 5-1 Nae Consruc a line parallel o he given line ha passes hrough a poin no on he line Given: Poin P is off a given line Task: Consruc a line hrough P parallel o he given line. 1. Wih your sraighedge, draw a ransversal hrough poin P. This is siply a sraigh line, which runs hrough P and inersecs he given line. 2. Using your knowledge of he consrucion COPY AN ANGLE, consruc a copy of he angle fored by he ransversal and he given line such ha he copy is locaed UP a poin P. The verex of your copied angle will be poin P. 3. When he copy of he angle is coplee, you will have wo parallel lines. This new line is parallel o he given line. P A B

20 Consruc a perpendicular o a line hrough a poin on he line Given: Poin P is on a given line Task: Consruc a line hrough P perpendicular o he given line. 1. Place your copass poin on P and sweep an arc of any size ha crosses he line wice (below he line). 2. STRETCH THE COMPASS LARGER!! 3. Place he copass poin where he arc crossed he line on one side and ake a sall arc below he line. 4. Wihou changing he span on he copass, place he copass poin where he arc crossed he line on he OTHER side and ake anoher arc. Your wo sall arcs should be crossing. 5. Wih your sraighedge, connec he inersecion of he wo sall arcs o poin P. This new line is perpendicular o he given line. P A B

21 Consruc a perpendicular o a line fro a poin off he line Given: Poin P is off a given line Task: Consruc a line hrough P perpendicular o he given line. 1. Place your copass poin on P and sweep an arc of any size ha crosses he line wice. 2. Place he copass poin where he arc crossed he line on one side and ake an arc ON THE OPPOSITE SIDE OF THE LINE. 3. Wihou changing he span on he copass, place he copass poin where he arc crossed he line on he OTHER side and ake anoher arc. Your wo new arcs should be crossing on he opposie side of he line. 4. Wih your sraighedge, connec he inersecion of he wo new arcs o poin P. This new line is perpendicular o he given line. P A B

22 Consrucions wih Parallel and Perpendicular Lines Geoery Uni 3: Lesson 5-2 Nae 1. Consruc an aliude fro B o AC. B A C 2. Consruc a square of lengh s. s

23 3. Draw a line l. Consruc a segen of lengh c ha is perpendicular o line l. c 4. Consruc a recangle wih base b and heigh a. a b

24 5. Consruc a recangle wih one side of lengh a and diagonal of lengh b. a b 6. Consruc a quadrilaeral wih a pair of opposie parallel sides of lengh c. c

25 Slope and he Equaions of Lines Geoery Uni 3: Lesson 6-1 Nae Slope Raio of he verical change (rise) o he horizonal change (run) beween any wo poins. Slope = y 2 y 1 x 2 x 1 Posiive slope Negaive slope Slope of verical line Slope of a horizonal line Line goes up fro lef Line goes down lef Undefined slope Zero slope o righ o righ y y y y x x x x 1. Find he slope of he given line. 2. Find he slope of he line passing hrough poins (-1, 2) and (7, -3). 3. Find he slope of he line passing hrough poins (1, 2) and (1, 6).

26 Equaions of a Line Slope Inercep For Poin Slope For Sandard For Y = x + b y y1 = (x x1) Ax + By = C Slope Y-Inercep Slope (x1, y1) is a poin on he line Use if you know he slope & y-inercep 1. Find slope Use he oher 2. Inpu slope for ehods for eq 3. Inpu he poin for x1 y1 Then ransfor 4. Disribue ino his for NO fracions For A, B, or C Equaions of horizonal lines have he for: y = # Equaions of verical lines have he for: x = # The y-coordinae never changes The x-coordinae never changes y x = -3 x = 0 x = 2 x = 5 y (-5, 5) (-2, 5) (1, 5) (4, 5) y = 5 (2, 5) y = 2 (2, 2) x y = 0 (2, 0) x y = -3 (2, -3) 4. Wrie he equaion of a line wih a slope of 3 and a y-inercep of -5.

27 5. Wrie he equaion of a line ha passes hrough (-1, 5) and has a slope of 2. Express in poin-slope for and slope-inercep for. 6. Wrie he equaion of a line ha passes hrough poins (-2, -1) and (3, 5). Express in poin-slope for and slope-inercep for.

28 7. Graph he equaion y x Graph he equaion y 3 = -2(x + 3). Does poin (9, 7) lie on he line? Does poin (4, 10) lie on he line? Jusify your answer. Jusify your answer. 9. Wrie he equaion of a verical line ha 10. Wrie he equaion of a horizonal line ha passes hrough poin (2, 4). passes hrough poin (4, -3).

29 Slopes of Parallel and Perpendicular Lines Geoery Uni 3: Lesson 6-2 Nae 1. Are lines a and b parallel? a b a) Find he slope of line a. b) Find he slope of line b. c) Copare he slopes of lines a and b. 2. Find he equaion of anoher line ha is parallel o lines a and b. Then graph i on he sae coordinae plane.

30 3. The equaions of wo lines are shown below. Are lines r and s perpendicular? Line r: 4y + 12 = x Line s: 8y + 2x = 16 a) Find he slope of line r. b) Find he slope of line s. c) Copare he slopes of lines r and s. 4. Deerine wheher he following lines are perpendicular. 3y 2x = 6 3x = 14 2y

31 5. Line p is represened by he equaion 2y + 2 = 6x. Find he following: Line n, a line ha is parallel o line p and ha passes hrough he poin (6, 2) Line q, a line ha is perpendicular o line p and ha passes hrough he poin (6, 2) a) Rewrie he equaion for p in slope-inercep for. Find he slope. b) Find he equaion of n. Wrie in slope-inercep for. c) Find he equaion of line q. Wrie in slope-inercep for. 6. Graph lines p, n, and q on he sae coordinae plane. Confir ha line p is parallel o line n and is perpendicular o line q.

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