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Transversal: Name the obvious transversal(s): c 1. a a 2. b 3. a c b c b When 2 coplanar lines are cut by a transversal, 8 angles are formed: INTERIOR ’s: EXTERIOR ’s: Some of these angles have a relationship that we have previously studied. LINEAR PAIRS: VERTICAL ’s: c b 6 7 a 5 8 1 2 3 4 1 Types of Angles Alternate Interior Angles: Alternate Exterior Angles: Same-side Interior Angles (consecutive angles): Corresponding Angles: Use the given line as a transversal: 1. Name alt. int ’s using line x: 2 2. Name s.s. int. ’s using line y: 1 3 4 3. Name corr. ’s using line z: 6 4. Name alt. ext. ’s using line y: 8 7 5. Name alt. int. ’s using line z: 5 10 9 y x z 6. Name s.s. int ’s using line z: 12 11 Examples – Find the measures of the angles (or value of the variables(s)). 1. 5. 3. 2. 6. 7. 4. 8. 2 9. 10. 11. 3 Is it possible to prove the lines are parallel or not parallel? If so, state the postulate or theorem you would use. If not, state cannot be determined. 1. 2. 3. 92° l k 88° 4. 5. 6. l k 7. A 105° E 8. B I C D 9. k 122° 75° H 55° l F 58° 55° m G Find the value of x so that n || m. State the theorem or postulate that justifies your solution. 5x 10. n 11. m 5x+23 n 7x+13 m 12. 5x-18 n m 8x-5 3x+48 x= x= x= _____________________ _____________________ _____________________ Can you prove that lines p and q are parallel? If so, state the theorem or postulate that you would use. p 13. p 14. q q _____________________ p 15. q _____________________ _____________________ 4 3.3 Parallel Proofs t l 1 2 m 3 4 5 1. Given: l // m; 1 4 Prove: s // t 1. l // m ; 1 4 1. _______________________________________ 2. 3 1 2. _______________________________________ 3. 3 4 3. _______________________________________ 4. s // t 4. _______________________________________ 2. Given: l // m ; 2 5 Prove: s // t 1. l // m ; 2 5 1. ________________________________________ 2. 2 3 2. ________________________________________ 3. 3 5 3. ________________________________________ 4. s // t 4. ________________________________________ 3. Given: l // m; s // t Prove: 2 4 1. l // m ; s // t 1. ________________________________________ 2. 2 3 3 4 3. 2 4 2. ________________________________________ 3. ________________________________________ 4. Given: l // m; s // t Prove: 1 5 1. l // m; s // t 1. _________________________________________ 2. 1 3 3 5 3. 1 5 2. _________________________________________ 3. _________________________________________ 5 5. Given: 3 is supplementary to 5. Prove: BD // FE 1. 3 is supplementary to 5 1. _________________________________________ 2. m3 + m5 = 180 2. _________________________________________ 3. 3 4 3. _________________________________________ 4. m3 = m4 4. _________________________________________ 5. m4 + m5 = 180 5. _________________________________________ 6. 4 is supplementary to 5 6. _________________________________________ 7. 7. _________________________________________ BD // FE B A 6. Given: 2 5; BE bisects CBD. Prove: AC // DE 1 D 1. 2 5; BE bisects CBD. 2. 3. 3 5 3. 4. 4. AC // DE 8. Given: l // m ; s // t Prove: 1 5 9. Given: l // m; 1 4 Prove: s // t C 3 4 5 E 1. 2. 3 2 7. Given: l // m ; s // t Prove: 2 4 2 s Diagram for # 7 - 10 l t 1 2 m 3 5 4 10. Given: l // m; 2 5 Prove: s // t 6 11. Given: BC // EF ; BA // ED Prove: B E A D C P B E 12. Given: AB //DE Prove: mACD = mBAC + mCDE F A B C E D 13. Given: g // h; g // j Prove: 2 3 g 1 2 h 3 14. Given: AB //CD ; BC //DE Prove: B D j C A E B D 15. Given: a // c; 1 2 Prove: b // c 2 1 a 16. Given: C is a supplement of D Prove: A is a supplement of B b B A c C D 7 3.4 Perpendicular Lines Perpendicular Bisector is a line perpendicular to a segment at the segment's midpoint. Distance from a point to a line the length of the perpendicular segment from the point to the line. 1- 6 Use the given diagram on the right, in which AM = MB. C 1. Name a pair of rays. 2. ____ is the bisector of ____. 3. Name a linear pair of angles which are . 4. If t in X is to 5. If A M B AB at M, what can you say about t and CM ? Why? MR in X is a bisector of AB , then R is on CM . Why? 6. If p contains M and is to the plane determined by CM and AB , then p ___ CM and p ___ AB . Why? 7. In a plane , how many lines can be to a given line at a given point? 8. Would your answer be different if the words “in a plane” were omitted from the question? Homework on Perpendicular Lines. True or False. If false, give a counterexample. _____1. If PQ PR , then QPR is a right angle. ______2. If AB CD , then ABC is a right angle. _____3. If 2 lines intersect to form a right angle, then the lines are . ______4. There is exactly one line to a given at a given point on the line. _____5. If 2 angles are a linear pair, then each is a right . ______6. A given segment has exactly one bisector. _____7. If M is the midpoint of AB and if AB is to plane X at M, there is exactly one line in X which ______8. If 2 adjacent angles are , then each is a right angle. is a bisector of AB . 8 In 9 – 13 refer to the diagram below and the given info. : mCAB = 90; CDA BDA; EA AB ; mECB = 90 C **Mark the diagram with the given information** X 9. What pairs of lines are ? D 10. ____ is a bisector of ____. Why? E A 11. If FC in X is a bisector of EB , then F is on AC . Why? 12. If t is a line in the plane of the diagram, and t BC at D, how are t and AD related? Why? 13. If G is on B CE and EGA CGA, how are AG and EC related? Why? 14. If l m and m n, is l n? Explain. 15. If m n, is n m ? Explain. For Exercises 3 and 4, name the shortest segment from the point to the line and write an inequality for x. 3. 4. ________________________________________ ________________________________________ 9 3.5-3.6 Slopes of Lines 10 Write the equation of each line in the given form. 1. the horizontal line through (3, 7) in 2. the line with slope point-slope form 8 through (1, 5) in 5 point-slope form 1 7 3. the line through , and (2, 14) in 2 2 slope-intercept form 4. the line with x-intercept 2 and y-intercept 1 in slope-intercept form Show All Work! Solve. 1. x2 – 81 = 0 2. x2 – 7x + 10 = 0 3. x2 + 10x + 25 = 0 4. 10x – 24 = x2 5. 6x2 – 5x + 1 = 0 6. x2 – 3x = 0 7. x2 – x – 12 = 0 8. x2 – 2x = 3 9. x2 + 49 = 14x 10. 2x2 + 7x = -6 11. 6x2 – x = 12 12. 10x2 + 3x = 18 In the problems, l // m. Find x. t 1 2 4 3 l 5 6 8 m 7 13. m4 = x2 + 72; m5 = -16x + 171 14. m3 = x2 – 2x ; m6 = 3x + 108 15. m3 = x2 + 2x ; m7 = -x + 70 16. m4 = 2x2 + 10; m6 = 11x – 5 1. Name the slope and y-intercept. x a) y 1 b) 3x + 2y = 6 4 c) y – 4 = 2 (x 2) 3 2. Write an equation of the line in Standard Form. a) m = -2 , b = 5 b) m = - 2 and the line contains (9,3) 5 11 c) the line contains (2,4) and (-3, 1) d) M(-2, 1) , N(2,4). Write the equation of the line // to MN with y-intercept of –4. e) Write an equation of the line through K(1,-2) to MN . f) A(1,-5) , B(-3, 5). Write an equation of the bisector of AB . 1. The vertices of a ∆ are A(-1,2), B(-1, 8) and C(3,5). Classify the ∆ by its sides. 2. The vertices of ∆JKL are J(-5,0), K(5,8) and L(4,-1). Is ∆JKL equilateral? 3. Find the perimeter of ∆GHI with vertices: G(8,5), H(-1,-4), and I(-4,0). Exact answer! 4. The vertices of ∆ABC are A(-2,-1), B(-6,7) and C(4,2). Prove that ∆ABC is a right ∆. 5. Find the slope of a line parallel to AB , given A(-5,4) and B(3,6). 6. Given: A(-4,-3), B(-2,-9), C(-4,1) and D(-7,0) Are AB and CD //, , or neither? 7. Write each equation in Standard Form: a) y = 3x + 6 b) y 3 5 ( x 3) 3 c) -y = 1 x 5 2 8. Write an equation of the line through the point (-4,5) // to y = 2x + 6 . 9. Write an equation of the line through the point (3,-1) to x – 3y = -2 10. Write the equation of the line through A(4,1) // to the line through B(-2,3) and C(-4,-3). 11. Write the equation of the bisector of AB , given A(5,2) and B(-1,4). 12. Find the point of intersection of 3x+y = 5 and the line containing (8,1) with a slope of 1 . 3 12