Parallel and Perpendicular Lines

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Transversal

A line that cuts across two or more (usually parallel) lines.

Transversal

A line that intersects 2 or more coplanar lines.

Transversal

A line that intersects two or more other lines at different locations.

Linear Pair

A pair of adjacent, supplementary angles.

Corresponding Angles

A pair of angles the correlate to the same spot but in a different intersection.

Vertical Angles

A pair of opposite congruent angles formed by intersecting lines

Linear Pair

Angle 1 and Angle 2

Vertical Angles

Angle 1 and Angle 4

Corresponding Angles

Angle 2 and Angle 6

Alternate Exterior Angles

Angle 2 and Angle 7

Same-Side Interior Angles

Angle 3 and Angle 5

Alternate Interior Angles

Angle 3 and Angle 6

corresponding angles

Angles in the same place on different lines

consecutive interior angles

Angles that are on the same side of a transversal on the interior of the parallel/intersected lines

Adjacent Angles

Angles that have a common side and a common vertex (corner point).

corresponding angles

Angles that have the same position on two different parallel lines cut by a transversal.

Alternate Interior Angles

Angles that lie inside the two lines and on the opposite sides of the transversals.

Consecutive Interior Angles

Angles that lie inside the two lines and on the same side of the transversal.

alternate exterior angles

Angles that lie outside a pair of lines and on opposite sides of a transversal.

Alternate Exterior Angles

Angles that lie outside the two lines and on the opposite sides of the transversals.

Consecutive Exterior Angles

Angles that lie outside the two lines and on the same side of the transversal.

Parallel because both lines have the same slope (-3/2) and different y-intercepts.

Are the lines parallel, perpendicular or neither? Explain how you know.

Parallel. They both have the same slope (2) and different y-intercepts

Are the lines parallel, perpendicular or neither? Explain how you know.

Perpendicular- because the slope of one line is 1/4 and the slope of the other line is -4

Are the lines parallel, perpendicular or neither? Explain how you know.

Standard Form

Ax + By = C

Parallel Lines

Coplanar lines that do not intersect.

Perpendicular

Determine if the line AC and line BD are parallel, perpendicular or neither. A(4,1), B(5,2) and C(4,2), D(5,1)

Neither

Determine if the line AC and line BD are parallel, perpendicular or neither. A(4,1), B(6,2) and C(4,3), D(5,1)

m = -7/6

Find the slope of a line that is perpendicular to the line that is created by the points (-2,1) and (5,7)

m = 1

Find the slope of line FG if F(1,1) and G(2,2)

m = 2/3

Find the slope of line FG if F(1,6) and G(-2,4)

-3

Find the slope of the line that is parallel to y=-3x-9

m = -1/5

Find the slope of the line that is perpendicular a line with the slope of 5

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then its on the perpendicular bisector on the segment

Alternate Exterior Angles Converse

If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

Alternate Interior Angles Converse

If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.

Consecutive Exterior Angles Converse

If two lines are cut by a transversal so the consecutive exterior angles are supplementary, then the lines are parallel.

Consecutive Interior Angles Converse

If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.

Corresponding Angles Converse

If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.

Consecutive Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pair of consecutive exterior angles are supplementary.

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pair of consecutive interior angles are supplementary.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior lines are congruent.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Parallel Lines

Lines that do not intersect and are coplanar

Skew Lines

Lines that do not intersect and are not coplanar

skew lines

Lines that do not intersect and are not coplanar

Skew Lines

Lines that do not intersect and are not coplanar.

Parallel Lines

Lines that do not intersect because they have the same slope.

Parallel Lines

Lines that do not intersect, but lie on the same plane.

Perpendicular Lines

Lines that form a right angle.

Parallel Planes

Planes that do not intersect.

Slope

Ratio of vertical change to horizontal change between any 2 points.

Slope

Rise over run The ratio of the rise and the run between any two points in a line

Parallel Slopes

Slopes of parallel lines are the same so that they never intersect.

Perpendicular Slopes

Slopes of perpendicular lines are opposite reciprocals of each other so that they intersect at 90 degrees.

Corresponding Angles

The angles that occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, these angles have equal measure.

Slope formula

The change in y over the change in x

Run

The difference in the x-values of two points on a line

Rise

The difference in the y-values of two points on a line

Point Slope Form of a Line

The equation of a line you can write when you were given a slope and a coordinate point the line must go through.

Slope Intercept Form of a Line

The graph-able form of a line that is set up for identifying the slope and the y-intercept

Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

slope

The steepness of a line on a graph

slope

The steepness of a line on a graph Rise over run

Triangle Sum Theorem

The sum of the measures of the angles of a triangle is 180.

Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is 180 degrees

Parallel Postulate

Through a point not on a line, there is one and only one line parallel to the given line.

Supplementary Angles

Two angles whose sum is 180 degrees

Complementary Angles

Two angles whose sum is 90 degrees

Alternate Exterior Angles

Two non adjacent angles on the exterior of the parallel lines and on alternate sides of the transversal.

Same Side Exterior Angles

Two non adjacent angles on the exterior of the parallel lines and on the same side of the transversal.

Alternate Interior Angles

Two non adjacent angles on the interior of the parallel lines and on alternate sides of the transversal.

Same Side Interior Angles

Two non adjacent angles on the interior of the parallel lines and on the same side of the transversal.

Parallel Planes

Two planes that do not intersect.

-3 because it needs to have the same slope.

What is the slope of a line parallel to this line?

1/2 because the slope of the given line is -2. The opposite reciprocal will make the line perpendicular because -2(1/2) = -1

What is the slope of a line perpendicular to this line?

converse

a conditional ("if-then") statement which reverses the order of the hypothesis ("if") and the conclusion ("then") parts of the statement.

transversal

a line that crosses 2 other lines; sometimes the other 2 lines are parallel

Transversal

a line that intersects two or more coplanar lines at different points

transversal

a line that intersects two or more lines

point-slope form

a linear equation of a non-vertical line

vertical angles

a pair of opposite congruent angles formed by intersecting lines

midpoint formula

a point that divides a segment into two congruent segments; the middle point of a segment; on the coordinate plane, calculated by the formula above

slope-intercept form

an equation written in the form y=mx+b is in slope-intercept form. The graph is a line with slope m and y-intercept b.

alternate interior angles

angles between 2 lines and on opposite sides of a transversal

Corresponding Angles

angles formed by a transversal that have corresponding positions

Consecutive Interior Angles

angles formed by a transversal that lie between the two lines and on the same side of the transversal

Alternate Interior Angles

angles formed by a transversal that lie between the two lines but on opposite sides of the transversal

Alternate Exterior Angles

angles formed by a transversal that lie outside the two lines but on opposite sides of the transversal

interior angles

angles on the inside of parallel lines cut by a transversal

exterior angles

angles on the outside of parallel lines cut by a transversal

Congruent Angles

angles that have the same measure

horizontal and vertical lines...

are always perpendicular

equidistant

at equal distances

Slope Formula

change in y over change in x

slope formula

change in y over change in x; rise over run

parallel lines

coplanar lines that do not intersect

congruent

equal

opposite reciprocals

example: 2/3 is -3/2

parallel lines

have slopes that are equal

perpendicular lines

have slopes that are negative (opposite) reciprocals

Perpendicular Bisector Theorem

if a point is on the perpendicular bisector of a segment then it is equidistant from the endpoints of the segment

Parallel Lines

lines in the same plane that never intersect

parallel lines

lines in the same plane that never intersect

Skew Lines

lines that do not intersect and are not coplanar

Coincidental Lines

lines that have equivalent linear equations and overlap at every point.

parallel lines

lines that have the same slope

Perpendicular Lines

lines that intersect to form right angles

perpendicular lines

lines whose slopes are opposite reciprocals; lines that intersect to form four 90-degree angles

Slopes of Perpendicular Lines are what?

opposite reciprocal

parallel planes

planes that do not intersect

slope

rate of change of a line

equidistant

the distance between two lines measured along a perpendicular line is always the same

perpendicular bisector

the equation of the line that goes through the MIDPOINT of a segment on the coordinate plane and has a slope that is the OPPOSITE RECIPROCAL of the original segment's slope

slope-intercept formula

the form of the equation of a line that gives you the slope and the y-intercept; can be found if you start with 2 points OR with a point and the slope

point-slope formula

the form of the equation of a line when given a point and the slope to start with

Exterior Angles Theorem

the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles

rate of change

the relationship between two quantities that are changing. The rate of change is also called slope. rate of change=change in the dependent variable/change in the independent variable.

Slopes of Parallel Lines are what?

the same

Vertical Angles

two nonadjacent, opposite angles formed by two intersecting lines

vertical lines have a slope that is...

undefined

corresponding angles

when 2 lines are crossed by a transversal: angles located in the same place relative to their intersection points; when the 2 lines are parallel, these angles are congruent, as shown in image

alternate interior angles

when 2 lines are crossed by a transversal: angles located on opposite sides of the transversal and between the other 2 lines; when the 2 lines are parallel, these angles are congruent, as shown in image

alternate exterior angles

when 2 lines are crossed by a transversal: angles located on opposite sides of the transversal and outside the other 2 lines; when the 2 lines are parallel, these angles are congruent, as shown in image

same-side interior angles

when 2 lines are crossed by a transversal: angles located on the same side of the transversal and between the other 2 lines; when the 2 lines are parallel, these angles are supplementary as shown in image; also known as consecutive interior angles

Undefined Slope

x = 2

vertical line

x = equation VUx

Point-Slope Form

y - y₁ = m (x - x₁)

Slope is zero

y = -3

horizontal line

y = equation H0y

Slope-Intercept Form

y = mx + b

point-slope form

y-y₁=m(x-x₁)

Point-Slope Formula

y=m(x-x-point) - y-point

slope-intercept form

y=mx+b

the slope of a line PERPENDICULAR to a line through (2,4) and (-1,-5)

-1/3

the slope of a line PERPENDICULAR to a line through (4,2) and (-5,-1)

-3

Equations of Parallel Lines

-same slope -different y-intercepts

Equations of Perpendicular Lines

-slopes are: negative reciprocals -form a right angle (90 degrees) at their intersection

Equations of Perpendicular Lines

-slopes are: opposite signs, reciprocals -form a right angle (90 degrees)

horizontal lines have a slope of...

0

the slope of a line PARALLEL to a line through (4,2) and (-5,-1)

1/3

the slope of the line through the points (3,5) and (-2,4)

1/5

the slope of a line PARALLEL to a line through (2,4) and (-1,-5)

3


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