Parallel and Perpendicular Lines
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