Chapter 3
Same-side Interior Angles Theorem
- if two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel
Two Lines Parallel to a Third Line
- if two lines are parallel to the same line, then all three lines are parallel to each other.
Perpendicular Lines Theorem 1
- if two lines are perpendicular, then they intersect to form right angles
Is the line with a slope of m = 7/6 upward or downward from left to right, or is horizontal or vertical?
Upward; right to left
Is the line with a slope of m = -3 upward or downward from left to right, or is horizontal or vertical?
Downward; left to right
Midpoint Definition
- Midpoint: of a line segment AB is the point that is equidistant from the endpoints A and B.
flow proof
- arrows show the logical connections between the statements - reasons are written below the statements
slope-intercept form of linear equation
- form of an equation of a non-vertical line is y = mx + b - If a linear equation is in the form y = mx + b, the slope of the line is b, the y-intercept point is (0,y), and the y-intercept is b.
point-slope form of linear equation
- form of an equation of a non-vertical line is y-y1 = m(x-x1), where m is the slope and (x1,y1) is the point on the line
Alternate Exterior Angles Theorem
- if two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel
Alternate Interior Angles Theorem
- if two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel
Perpendicular Transversal Theorem
- in a plane, let two parallel lines be cut by a transversal. - If the transversal is perpendicular to one of the parallel lines, then it is perpendicular to the other parallel line
corresponding angles
- lie on the same side of the transversal and in corresponding positions
alternate interior angles
- non-adjacent interior angles that lie on opposite sides of the transversal - 5 and 6 - 7 and 8
slope or rate of change
- tilt or steepness of a line - RISE over RUN - the slope of m of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points
Slopes of Perpendicular Lines
- two intersecting lines with right angles - the rise and run switched, the run now becomes negative
perpendicular lines
- two lines intersecting at right angles - two non-vertical lines are perpendicular if the product of their slopes is -1
parallel segments
- two segments are parallel if the lines containing them are parallel
vertical line - linear equation
- x = c - The slope is undefined, and the x-intercept point is (c,0)
horizontal line - linear equation
- y = c - the slope is 0, and the y-intercept point is (0,c)
Defining: Alternate, interior, exterior and corresponding
- Alternate: lie on alternate sides of transversal line - Interior: lie in the interior of two lines with a transversal line - Exterior: line on the exterior of two lines with a transversal line - Corresponding: lie in corresponding (or same) positions with respect to the transversal line, except one angle is formed with each line
Standard form of linear equation
- Ax + By = C - where A and B are not both 0
Complementary Angles
- Complementary angles are two angles with a sum of 90∘ - A common case is when they FORM A RIGHT angle.
Midpoint Formula
- Distance formula: Suppose that A = (x1, y1) and B = (x2, y2) are two points in the coordinate plane.
Perpendicular Lines Theorem (Linear pair)
- If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular to each other
Rectangular Coordinate System
- Rectangular Coordinate System: consists of a vertical line called the y−axis and a horizontal line called the x−axis. The x−axis and y−axis divide the coordinate plane into four quadrants and intersect at a point called the origin. Each point in the plane corresponds to a unique ordered pair (x, y).
Supplementary Angles
- Supplementary angles are two angles with a sum of 180∘ - A common case is when they lie on the SAME SIDE of a straight line
y-intercept point
- a line is the point where the line crosses the y-axis, so has the form (0,y)
transversal
- a line that intersects two or more coplanar lines at different points - transversal has nothing to do with whether lines are parallel or not
interior angles
- an angle formed between parallel lines by a third line that intersects them
exterior angles
- an angle formed outside parallel lines by a third line that intersects them
skew lines
- are non-coplanar - they are not parallel and do not intersect
Slopes of Parallel Lines
- have the same slope and different y-intercepts
Corresponding Angles Theorem
- if two lines and a transversal form corresponding angles that are congruent, then the lines are parallel
Alternate Exterior Angles Converse
- if two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Alternate Interior Angles Converse
- if two parallel lines are cut by a transversal, then corresponding angles are congruent
Corresponding Angles Converse
- if two parallel lines are cut by a transversal, then corresponding angles are congruent
Same-Side Interior (or Consecutive) Angles Converse
- if two parallel lines are cut by a transversal, then same-side interior angles are supplementary
Two Lines Perpendicular to the Third Line Theorem
- in a plane, if two lines are perpendicular to the same line, then they are parallel to each other
same-side interior angles or consecutive interior angles
- interior angles that lie on the same side of the transversal (sometimes called consecutive interior angles) - 3 and 5 - 4 and 6
alternate exterior angles
- non-adjacent exterior angles that lie on opposite sides of the transversal
parallel lines
- parallel lines are coplanar lines that do not intersect - the symbol II means - is parallel to - two non-vertical lines are parallel if they have the same slope and different y-intercepts
parallel planes
- planes that do not intersect
vertical change
- rise
horizontal change
- run
slope-intercept form of linear equation definitions
- slope-intercept form means the equation is solving for y - if two points of a line are given, either one may be used to write an equation of the line - Remember that only when an equation is solved for y is the coefficient of x the slope - for a line with positive slope m, as m increases, the line becomes steeper
slope
- the slope of any vertical line is undefined - the slope of horizontal line is 0
y-intercept
- the y coordinate only of this point, or simply y
Parallel Postulate
- through a point not on a line, there is one and only one line parallel to the given line - There is exactly one line through P parallel to l
Perpendicular Postulate
- through a point not on a line, there is one and only one line perpendicular to the given line - There is exactly one line through P perpendicular to l
Common Mistakes to Avoid with midpoints and distance
1. The midpoint is found by averaging the x−coordinates and averaging the y−coordinates. Do NOT subtract them. 2. The square root of a sum is NOT the sum of the square roots 3. When using the Pythagorean Theorem, make sure that the hypotenuse is on a side by itself; namely, leg2 + leg2 = hypotenuse2.
Is the line with a slope of m = 0 upward or downward from left to right, or is horizontal or vertical?
Horizontal
Is the line with a slope of m = undefined upward or downward from left to right, or is horizontal or vertical?
Vertical