Chapter 3 Midterm Review

1. Chapter 3 Midterm Review Parallel and Perpendicular Lines By: James, Jeremy, Eric, and Karthik

2. Introduction • Chapter 3 is about parallel and perpendicular lines. Parallel and perpendicular line theorems are very important. • Section 1-3: Identifying angle relations given parallel lines with transversals. Incorporate angle relationships into proofs. Use algebra to find angle measures. • Section 3-4: Using slope equations to identify parallel and perpendicular lines. • Section 5: Using angle relationships to prove lines parallel. • Section 6: Finding the distance between a point and a line as well a between two parallel lines.

3. Section 1: Key Terms • Parallel Lines: Lines or segments that do not intersect with one another. Parallel lines are usually mark with corresponding arrows. • Parallel Planes: Like parallel lines, planes can be parallel to one another • Skew Lines: Lines that do not intersect and are not coplanar • Transversal: A line that intersects two or more lines in a plane at different points g Lines e and f are parallel. Line g is the transversal. e f

4. Section 1: Angle Pair Relationships • Corresponding angles are angles in the same place on two sides of a transversal. • Alternate interior angles are on alternate sides of the inside of the transversal. • Alternate exterior angles are on alternate sides of the outside of the transversal. • Consecutive interior angles are consecutive angles on the inside of the transversal.

5. Section 1 Example • Angle 3 and Angle 5 are alternate interior. • Angles 2 and 7 are alternate exterior. • Angles 4 and 5 are consecutive interior. 1 2 3 4 5 6 7 8

6. Section 2: Angles and Parallel Lines • Corresponding Angles Postulate (CAP): When two parallel lines are cut by a transversal, the corresponding angles are congruent. c Line a parallel to line b <1 ≡ <5, <2 ≡ <6 <7 ≡ <3 , <8 ≡ <4 a 1 2 3 4 b 5 6 7 8

7. Section 2 Continued • Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, each pair of alternate interior angles is congruent • Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, each pair of consecutive interior angles are supplementary • Alternate Exterior Angles: If two parallel lines are cut by a transversal, the alternate exterior angles are congruent

8. Section 3: Slopes of Lines • Words to Know: • Slope: the rise and run of a line. • Formula: Delta y over delta x • Commonly referred to with Cartesian Coordinates. • Rate of Change: how a quantity changes over time. • Special Slopes: • Parallel Lines: will have the same slope. • Perpendicular Lines: will have opposite reciprocals as their slopes.

9. Section 3 Example • Find the slope of this line

10. Math History • Rene Descartes – French Mathematician who invented the Cartesian Coordinate system and is widely recognized as the father of analytical geometry • Cartesian Coordinates: Specifies a point in a plane with horizontal and vertical coordinates (x,y). This can be expanded into higher dimensions with simply adding more variables (x,y,z, etc.) • Analytical Geometry: Also known as coordinate geometry, deals with geometry on the coordinate plane. It uses algebraic principles to solve geometric problems.

11. How Math History relates to this Chapter • Analytic Geometry, which Descartes developed is very closely tied to this chapter • Used to find slopes of lines in cartesian coordinates, along with their parallels and perpendiculars. • Used to find distances and midpoints between points • Used for finding the distance from a point to a line (perpendicular distance) as well as the distance between 2 parallel lines

12. Section 4: Equations of Lines • Equations of lines: • Point-slope form: y-y1=m(x-x1) • Slope-intercept form: y=mx+b • Uses for equations: • One slope, one point: Point-slope form • Two points: Slope -intercept form • One point, one equation: slope-intercept form

13. Section 4 Example • Write the equation of a line with point A(1,6) and the slope of -6 point slope form • What is the equation of a line containing the points (1,9) and (-8, 7)?

14. Section 5: Proving Lines Parallel Postulates: • CAP Converse: If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. • Parallel Lines Postulate: If given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.

15. Section 5: Continued • Theorems: • AEA Converse: If two lines are cut by a transversal so that the alternate exterior angles are congruent, then the two lines are parallel. • AIA Converse: If two lines are cut by a transversal so that the alternate interior angles are congruent, then the two lines are parallel. • CIA Converse: If two lines are cut by a transversal so that the consecutive interior angles are supplementary, then the lines are parallel. • Parallel Perpendicular Theorem: If two lines are perpendicular to the same line, then they are parallel.

16. Section 5 Example • If angles 1 and 7 are If angles 1 and 6 are congruent, congruent, then the lines are then the lines are parallel. congruent. If angles 3 and 7 are congruent, then the If the two lines are both perpendicular to lines are parallel. the transversal, they are parallel. 9 10 1 2 3 4 11 12 13 14 8 7 5 6 15 16

17. Section 6: Perpendicular Lines and Distance • Distance between a point and a line is the length of the segment perpendicular to the line from the point. • The distance between parallel lines is the distance from one point on one line to another point on the other line. • Theorem 3.9: In a plane, if two lines are equidistant from a third line, then the two lines are parallel.

18. Section 6 Example 1 • Find the distance between two lines with the equations of y = 6x – 9 and y = 6x + 3

19. Section 6 Example 2 • Find the distance between point A(-3, -4) and a line with the equation of y = -4x + 9

20. Questions?