Unit 3 - Parallel and Perpendicular Lines

addition property of equality

If a=b, then a+c=b+c

subtraction property of equality

If a=b, then a-c=b-c

Multiplication property of equality

If a=b, then axc=bxc

Division property of equality

If a=b, then a/c=b/c

Distributive property of equality

a(b+c), then a(b+c) = ab + ac

Substitution property of equality

If a=b, then a may be replaced by b in any expression or equation

reflexive property of equality

for any real number a, a=a

Symmetric property of equality

if a=b, b=a

transitive property

if a=b, and b=c, then a=c

reflexive property of congruence

For any segment AB, line AB is congruent to line AB

Symmetric property of congruence

If line AB is congruent to line CD, then line CD is congruent to line AB

transitive property of congruence

If line AB is congruent CD and line CD is congruent to line EF, then line AB is congruent to line EF

Definition of congruence

Segments are congruent if and only if they have the same measure:
If line AB is congruent to line CD, then AB = CD
If AB = CD, then line AB is congruent to line CD

Definition of midpoint

If M is the midpoint of line AB, then line AM is congruent to line MB

Segment Addition Postulate

If A, B, and C are collinear points and B is between A and C, then AB + BC = AC

Definition of congruence

The measures of two angles are equal if and only if the angles are congruent. If the measure of A = the measure of B, then angle A is congruent to angle B, vice versa.

Definition of a Right Angle

An angle measures 90 degrees if and only if it is a right angle. If the measure of A = 90 degrees, then angle A is a right angle

Definition of complementary angles

Two angles are complementary, if and only the sum of their measures is 90 degrees

Definition of supplementary angles

Two angles are supplementary, if and only the sum of their measures is 180 degrees

Definition of an angle bisector

An angle bisector divides an angle into two equal parts

Definition of perpendicular

perpendicular lines form right angles

Angle addition postulate

The measure of ABD and the measure of DBC = the measure of ABC

Vertical angles theorem

if two angles are vertical, then they are congruent.

complement theorem

If two angles form a right angle, then they are complementary

Linear pair theorem (supplement theorem)

If two angles form a linear pair, then they are supplementary

Congruent complements theorem

If two angles are complementary to the same angle, then they are congruent (if angle A is complementary to angle B, and angle C is complementary to angle B, then angle A is congruent to angle C)

Congruent supplements theorem

If two angles are supplementary to the same angle, then they are congruent (if angle A is supplementary to angle B, and angle C is supplementary to angle B, then angle A is congruent to angle C)

parallel lines

Two coplanar lines that never intersect

Parallel planes

Planes that never intersect

Skew lines

non-coplanar lines that never intersect

corresponding angles

Angles on the same side of the transversal and in the same position. When parallel lines are cut by a transversal, they are congruent

Alternate interior angles

Interior angles, non-adjacent, and on opposite sides of the transversal. When parallel lines are cut by a transversal, they are congruent.

Alternate exterior angles

Exterior angles, non-adjacent, and on opposite sides of the transversal. When parallel lines are cut by a transversal, they are congruent.

Consecutive (Same-side) interior angles

Interior angles that are on the same side of the transversal. When parallel lines are cut by a transversal, they are supplementary.

Consecutive (Same-side) exterior angles

Exterior angles that are on the same side of the transversal. When parallel lines are cut by a transversal, they are supplementary.

slope formula

m = y₂-y₁/x₂-x₁. (x₁, y₁) represents the first point. (y₂, x₂) represents the second point.

Standard form

ax + by = c (a represents the x-intercept, b represents the y-intercept)

slope-intercept form

y = mx + b (m represents the slope, b represents the y-intercept)

point-slope formula

y-y₁ = m(x-x₁). (x₁, y₁) represents the coordinates of a given point, m represents the slope.

the slopes of parallel lines are ____

equal

the slope of perpendicular lines are _______

negative reciprocals

transitive property of parallel lines

If two lines are parallel to the same line, then they are parallel to each other