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