1. Lesson 2-3: Pairs of Lines

2. Lesson 2-3: Pairs of Lines
Parallel Lines
Parallel lines are coplanar lines that do not intersect.
Arrows are used to indicate lines are parallel.
The symbol used for parallel lines is ||.
In the above figure, the arrows show that line AB is parallel to line CD.
With symbols we denote,
Lesson 2-3: Pairs of Lines

3. Lesson 2-3: Pairs of Lines
PERPENDICULAR LINES
Perpendicular lines are lines that intersect to form a right angle.
The symbol used for perpendicular lines is  .
4 right angles are formed. m n
In this figure line m is perpendicular to line n.
With symbols we denote, m  n
Lesson 2-3: Pairs of Lines

4. Skew Lines and Parallel Planes
Two lines are skew if they do not intersect and are not in the same plane (not coplanar).
Ex:
All planes are either parallel or intersecting. Parallel planes are two planes that do not intersect.
Ex: Plane ABC and Plane EFG
Lesson 2-3: Pairs of Lines

5. Lesson 2-3: Pairs of Lines
Examples:
Name all segments that are parallel to
Name all segments that intersect
Name all segments that are skew to
Name all planes that are parallel to plane ABC.
Answers:
Segments BC, FG, & EH.
Segments DH, DC, AE & AB.
Segments CG, BF, FE, & GH.
Plane FGH.
Lesson 2-3: Pairs of Lines

6. Lesson 2-4: Angles and Parallel Lines
Transversal
Definition: A line that intersects two or more lines in a plane at different points is called a transversal.
When a transversal t intersects line n and m, eight angles of the following types are formed:
Exterior angles
Interior angles
Consecutive interior angles
Alternative exterior angles
Alternative interior angles
Corresponding angles t m n
Lesson 2-4: Angles and Parallel Lines

7. Vertical Angles & Linear Pair
Two angles that are opposite angles. Vertical angles are congruent.
 1   4,  2   3,  5   8,  6   7
Supplementary angles that form a line (sum = 180)
1 & 2 , 2 & 4 , 4 &3, 3 & 1,
5 & 6, 6 & 8, 8 & 7, 7 & 5 1 2 3 4 5 6 7 8
Lesson 2-4: Angles and Parallel Lines

8. Corresponding Angles & Consecutive Angles
Corresponding Angles: Two angles that occupy corresponding positions.
 2   6,  1   5,  3   7,  4   8 1 2 3 4 5 6 7 8
Lesson 2-4: Angles and Parallel Lines

9. Lesson 2-4: Angles and Parallel Lines
Consecutive Angles
Consecutive Interior Angles: Two angles that lie between parallel lines on the same sides of the transversal.
Consecutive Exterior Angles: Two angles that lie outside parallel lines on the same sides of the transversal.
m3 +m5 = 180º, m4 +m6 = 180º 1 2
m1 +m7 = 180º, m2 +m8 = 180º 3 4 5 6 7 8
Lesson 2-4: Angles and Parallel Lines

10. Lesson 2-4: Angles and Parallel Lines
Alternate Angles
Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair).
Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal.
 3   6,  4   5
 2   7,  1   8 1 2 3 4 5 6 7 8
Lesson 2-4: Angles and Parallel Lines

11. Lesson 2-3: Pairs of Lines

12. Slope of Parallel and Perpendicular lines
The slope of the non vertical line through the points and is
m =
The slope of a vertical line is not defined.
The slope of a horizontal line is zero.
Two lines are parallel if and only if they have equal slopes.
Two lines are perpendicular if and only if the product of their slopes is -1 (reciprocals and opposite signs).
Lesson 2-3: Pairs of Lines

13. Lesson 2-3: Pairs of Lines
Examples:
Find the slope of the line through the given points.
(-4, 7) and (3, 7)
(3, -1) and (3, 2)
(1, -4) and (2, 5)
(-2, 5) and (1, -1)
Lesson 2-3: Pairs of Lines

14. Lesson 2-3: Pairs of Lines
Examples
Any line parallel to a line with slope has slope _____.
Any line perpendicular to a line with slope has slope ___.
Any line parallel to a line with slope 0 has slope _____.
Any line perpendicular to a line with undefined slope has slope.
Any line parallel to a line with slope 2 has slope _____.
Zero Slope 2
Lesson 2-3: Pairs of Lines