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Understanding Parallel Lines and Transversals in Geometry

Explore definitions of parallel lines, skew lines, and transversals in geometry. Learn about different types of angles formed by transversals and how they relate to each other. Practice angle measurements with examples.

shaeleigh-kemp
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Understanding Parallel Lines and Transversals in Geometry

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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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