Geometry. 3.1 Pairs of Lines & Angles 3.2 Parallel Lines and Transversals

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1 3.1 Pairs of Lines and Angles

2 Geometry 3.1 Pairs of Lines & Angles

3 Essential Question What does it mean when two line are parallel, intersecting, coincident, skew, or perpendicular? And what are the properties of angles formed by parallel lines cut with a transversal? 3.1 Pairs of Lines and Angles

4 Parallel Lines Coplanar lines that do not intersect. m n m n Small arrows are used in a diagram to show lines are parallel. 3.1 Pairs of Lines and Angles

5 Skew Lines Lines that do not intersect and are not coplanar. s r 3.1 Pairs of Lines and Angles

6 Parallel Planes Planes that don t intersect. 3.1 Pairs of Lines and Angles

7 Segments and Rays can be parallel. Sketch the following examples. B D AB CD A C MN OP O M P N 3.1 Pairs of Lines and Angles

8 Visualization AB and ED Parallel A B E D AB AB and EF Skew and BD Perpendicular Think of a rectangular box. F G 3.1 Pairs of Lines and Angles

9 Example 1 Think of each segment in the figure are part of a line. Identify each pair of lines as parallel, skew or perpendicular. E F Parallel A B Perpendicular Perpendicular Skew D C G 3.1 Pairs of Lines and Angles

10 Your turn Q L M P N Name a... Line parallel to Line perpendicular to Line skew to Plane parallel to plane RPL. R Plane SNM S 3.1 Pairs of Lines and Angles

11 Postulate 3.1 Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. 3.1 Pairs of Lines and Angles

12 Postulate 3.2 Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. 3.1 Pairs of Lines and Angles

13 Example 2 The given line markings show how the roads in a town are related to one another. Name a pair of parallel lines. DM and FE Name a pair of perpendicular lines. DM and BF Is No! 3.1 Pairs of Lines and Angles

14 Transversals A transversal cuts across two parallel lines at an angle. The transversal intersects the two lines at two different points. 3.1 Pairs of Lines and Angles

15 This is not a transversal. The lines intersect at only one point. 3.1 Pairs of Lines and Angles

16 Theorem 3.1:Corresponding Angles m n m n If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

17 This means ALL corresponding angles are congruent

18 Example 1 Find all angle measures in the picture. 120? 60? 120? 60??60?120 60?120 Notice When two parallel lines are cut by a transversal, any pair of angles will either be congruent or. supplementary

19 Theorem 3.2 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. 2 lines alt int s

20 Theorem 3.3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. 2 lines alt ext s

21 Theorem 3.4 Same Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of same side interior angles are supplementary. m 1 + m 2 = 180 m 3 + m 4 = lines ss int s supp

22 Theorems in a nutshell. 2 lines corr s

23 Theorems in a nutshell. 2 lines alt. int. s

24 Theorems in a nutshell. 2 lines alt. ext. s

25 Theorems in a nutshell. 2 lines ss int. s supp.

26 These are the reasons for proof. 2 lines corr s alt int s alt ext s ss int s supp

27 Example 2 State the theorem that justifies the statement below Alt Int s Alt Ext s Corr. s m 6 + m 7 = 180 Same Side Int s

28 Example 3 Solve each problem for x and y. Identify the theorem that justifies your answer. a. b. x y 70 y x 120

29 Example 4 (120 x) m m n Solve for x. 5x n 2 lines alt ext s 5x = 120 x 6x = 120 x = 20

30 Example 5 (x + 20) m m n Solve for x. (x + 8) n 2 lines SS int s supp (x + 20) + (x + 8) = 180 2x + 28 = 180 2x = 152 x = 76

31 Example 6 m n Solve for x. (x + 40) (x + 40) (x + 50) 2 lines corr s m Linear Pair Post. n (x + 40) + (x + 50) = 180 2x + 90 = 180 2x = 90 x = 45

32 In Summary. 2 lines corr s alt int s alt ext s ss int s supp

33 Extra Practice In each of the following problems solve for x and y. a. (2x - 10) b y x

34 Extra Practice In each of the following problems solve for x and y. c. d. 130 x y 120 3x

35 Extra Practice Solve for x and y. 5x = 35

36 Extra Practice In each of the following problems solve for x and y. e.

37 Extra for Experts Find x. (Hint: Draw a line through the vertex of angle x and parallel to the other two lines.) 25 x 43 m n m n

38 Solution x Find x. m n m n x = = 68

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