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3.3 ll Lines & Transversals

3.3 ll Lines & Transversals. p. 143. Definitions. Parallel lines (ll) – lines that are coplanar & do not intersect. (Have the some slope. ) l ll m Skew lines – lines that are not coplanar & do not intersect. Parallel planes – 2 planes that do not intersect.

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3.3 ll Lines & Transversals

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  1. 3.3 ll Lines & Transversals p. 143

  2. Definitions • Parallel lines (ll) – lines that are coplanar & do not intersect. (Have the some slope. ) lll m • Skew lines – lines that are not coplanar & do not intersect. • Parallel planes – 2 planes that do not intersect. Example: the floor & the ceiling l m

  3. 1. Line AB & Line DC are _______. 2. Line AB & Line EF are _______. 3. Line BC & Line AH are _______. 4. Plane ABC & plane HGF are _______. A B D C • Parallel • Parallel • Skew • Parallel H G E F

  4. Transversal • A line that intersects 2 or more coplanar lines at different points. t l m

  5. 2 • 3 4 5 6 7 8 Interior <s - <3, <4, <5, <6 (inside l & m) Exterior <s - <1, <2, <7, <8 (outside l & m) Alternate Interior <s - <3 & <6, <4 & <5 (alternate –opposite sides of the transversal) Alternate Exterior <s - <1 & <8, <2 & <7 Consecutive Interior <s - <3 & <5, <4 & <6 (consecutive – same side of transversal) Corresponding <s - <1 & <5, <2 & <6, <3 & <7, <4 & <8 (same location)

  6. Post. 15 – Corresponding s Post. • If 2  lines are cut by a transversal, then the pairs of corresponding s are . • i.e. If l m, then 12. 1 2 l m

  7. Thm 3.4 – Alt. Int. s Thm. • If 2  lines are cut by a transversal, then the pairs of alternate interior s are . • i.e. If l m, then 12. 1 2 l m

  8. Proof of Alt. Int. s Thm. Statements • l m • 3  2 • 1  3 • 1  2 Reasons • Given • Corresponding s post. • Vert. s Thm •   is transitive 3 1 2 l m

  9. Thm 3.5 – Consecutive Int. s thm • If 2  lines are cut by a transversal, then the pairs of consecutive int. s are supplementary. • i.e. If l m, then 1 & 2 are supp. l m 1 2

  10. Thm 3.6 – Alt. Ext. s Thm. • If 2  lines are cut by a transversal, then the pairs of alternate exterior s are . • i.e. If l m, then 12. l m 1 2

  11. Thm 3.7 -  Transversal Thm. • If a transversal is  to one of 2  lines, then it is  to the other. • i.e. If l m, & t  l, then t m. ** 1 & 2 added for proof purposes. t 1 2 l m

  12. Proof of  transversal thm Statements • l m, t  l • 12 • m1=m2 • 1 is a rt.  • m1=90o • 90o=m2 • 2 is a rt.  • t m Reasons • Given • Corresp. s post. • Def of  s • Def of  lines • Def of rt.  • Substitution prop = • Def of rt.  • Def of  lines

  13. 1 Ex: Find: m1= m2= m3= m4= m5= m6= x= 125o 2 3 5 4 6 x+15o

  14. Check It Out! Example 1 Find mQRS. x = 118 Corr. s Post. mQRS + x = 180° Def. of Linear Pair Subtract x from both sides. mQRS = 180° – x = 180° – 118° Substitute 118° for x. = 62°

  15. Example 2: Finding Angle Measures Find each angle measure. A. mEDG mEDG = 75° Alt. Ext. s Thm. B. mBDG x – 30° = 75° Alt. Ext. s Thm. x = 105 Add 30 to both sides. mBDG = 105°

  16. Assignment

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