angle addition postulate
if point B lies in the interior of ?AOC, then ?AOB + ?BOC = ?AOC
midpoint theorem
If M is the midpoint of line segment AB, then AM?�AB and MB?�AB
angle bisector theorem
If ray BX is the bisector of ?ABC, then ?ABX?�?ABC and ?XBC?�?ABC
vertical angles are congruent
If ?1 and ?2 are vertical ?s, then ?1??2
if two lines are perpendicular, then they form congruent adjacent angles
If line AC is perpendicular to line BD at point O, then ?AOB??BOC and ?AOB is adjacent to ?BOC
if two lines form congruent adjacent angles, then the lines are perpendicular
if ?AOB??BOC and ?AOB is adjacent to ?BOC, then line AC is perpendicular to line BD at point O
if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complimentary
if ?AOB and ?BOC are adjacent acute ?s and line A is perpendicular to line C, then ?AOB and ?BOC are complimentary
if two angles are supplements of two congruent angles, then the two angles are congruent
If ?1 and ?2 are supplementary and ?3 and ?4 are supplementary and ?1??3, then ?2??4
if two angles are compliments of two congruent angles, then the two angles are congruent
If ?1 and ?2 are complimentary and ?3 and ?4 are complimentary and ?1??3, then ?2??4
if two parallel planes are cut by a third plane, then the lines of intersection are parallel
If plane X is parallel to plane Y and plane Z intersects plane X at line L and plane Y at line M, then line L is parallel to line M
if two parallel lines are cut by a transversal, then the corresponding angles are congruent
If line L is parallel to line O and both are cut by transversal A, then the corresponding ?s are ?
if two parallel lines are cut by a transversal, then the alternating interior angles are congruent
If line L is parallel to line O and both are cut by transversal A, then the alt. int. ?s are ?
if two parallel lines are cut by a transversal, then the same-side interior angles are supplementary
If line L is parallel to line O and both are cut by transversal A, then the same-side interior angles are supplementary
in a plane, two lines perpendicular to the same line are parallel
if line A and line B are both ? to line X, then line A is || to line B
through a point outside a line there is exactly one line parallel to the given line
If point P and line A line in a plane, there is exactly one line through point P that is || to line A
two lines parallel to a third line are parallel to each other
If line Z is parallel to line O and line E is parallel to line O, then line Z is parallel to line E
the sum of the measures of the angles of a triangle is 180 degrees
Given: ?ABC
Conclude: ?A + ?B + ?C = 180�
the measure of an exterior angle of a triangle of a triangle is equal to the sum of the measures of the two remote interior angles
Exterior angle theorem (pretty sure)
the sum of the measure of the interior angles of a convex polygon with n sides is (n-2)180
Given: polygon M is a convex pentagon
Conclude: the sum of the interior angles of polygon M is 540�
the sum of the measures of the exterior angles of any convex polygon , one angle at each vertex, is 360 degrees
Given: polygon H is a convex hexagon
Conclude: the sum of the measures of the exterior angles of polygon H is 360�
SSS postulate
if three sides of one ? are congruent to three sides of another ?, then the ?s are ?
SAS postulate
if two sides and the included ? of one ? are congruent to two sides and the included ? of another ?, then the ?s are ?
ASA postulate
if two ?s and the included side of one ? are congruent to two ?s and the included side of another ?, then the ?s are ?
CPCTC
Given: ?ABC ? ?DEF
Conclude: ?A??D
isosceles triangle theorem
if two sides of a ? are ?, then the ?s opposite those sides are ?
converse of the isosceles triangle theorem
if two ?s of a ? are ?, then the sides opposite those ?s are ?
AAS postulate
if two ?s and the non-included side of one ? are ? to two ?s and the non-included side of another ?, then the triangles are ?
HL theorem
if the hypotenuse and a leg of one right ? are ? to the corresponding parts of another right ?, then the triangles are ?
If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment
if point C lies on the ? bisector of line segment JK, then point C is equidistant from point J and K
if a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment
if point A is equidistant from the endpoints of line segment DJ, then point a lies on the ? bisector of segment DJ.
if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
if point K lies on the bisector of ?AGJ, then point K is equidistant from lines A and J.
if a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle
if point S is equidistant from the sides of ?BHL, then point S lies on the bisector of ?BHL.
opposite sides of a parallelogram are congruent
Given: parallelogram WXYZ
Conclude: WX?YZ and XY?ZW
opposite angles of a parallelogram are congruent
Given: parallelogram WXYZ
Conclude: ?W??Y and ?X??Z
diagonals of a parallelogram bisect each other
Given: parallelogram WXYZ, WY and XZ intersect at point Q
Conclude: WQ?QY and XQ?QZ
if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Given: quad. ABCD, AB?CD and AD?BC
Conclude: ABCD is a parallelogram
if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram
Given: quad. ABCD, AB?CD, AB||CD
Conclude: ABCD is a parallelogram
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Given: quad. ABCD, ?A??C, ?B??D
Conclude: ABCD is a parallelogram
if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
Given: quad. ABCD, AC and BD intersect at point O, AO?OC, BO?OD
Conclude: ABCD is a parallelogram
if both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram
Given: quad. ABCD, AB||CD, AD||BC
Conclude: ABCD is a parallelogram
if two lines are parallel, then all points one one line are equidistant from the other line
If line A is || to line B, then all points on line A are equidistant from line B
if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
If line A || line B || line C and those lines cut off ? segments on transversal D, then they cut of ? segments on trans. E
a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the thrid side
Given: ?ABC, DE||BC, D is the midpoint of AB
Conclude: E is the midpoint of CA,
the segment that joins the midpoint of two sides of a triangle is parallel to the third side
Given: ?MRJ, A is the midpoint of MR, B is the midpoint of JM
Conclude: AB || RJ
the segment that joins the midpoint of two sides of a triangle is half as long as the thrid side
Given: ?MRJ, A is the midpoint of MR, B is the midpoint of JM
Conclude: AB?�RJ
the diagonals of a rectangle are congruent
Given: rectangle PQRS
Conclude: PR?QS
the diagonals of a rhombus are perpendicular
Given: rhombus PQRS
Conclude: PR?QS
each diagonal of a rhombus bisects two angles of the rhombus
Given: rhombus PQRS
Conclude: PR bisects ?P and ?R, QS bisects ?Q and ?S
the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
Given: right ?LMN, O is the midpoint of LN
Conclude: OL?ON?OM
if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle
Given: parallelogram ABCD, ?A?90�
Conclude: ABCD is a rectangle
if two consecutive sides of a parallelogram are conrgruent, then the parallelogram is a rhombus
Given: parallelogram STUV, ST?TU
Conclude: STUV is a rhombus
base angles of an isosceles trapezoid are congruent
Given: isosceles trapezoid MNOP
Conclude: ?M??P, ?N??O
the median of a trapezoid is parallel to the bases
Given: trapezoid ABCD, median XY
Conclude: XY||AD||BC
the median of a trapezoid has a length equal to the average of the base lengths
Given: trapezoid ABCD, median XY
Conclude: XY?AD+BC/2
exterior angle inequality theorem
the measure of an exterior ? of a ? is > the measure of either remote interior ?
if one side of a triangle is longer than a second side, then the opposite angle of the first side is larger than the opposite angle of the second side
Given: ?ABC, AB>BC
Conclude: ?C>?A
if one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle
Given: ?ABC, ?C>?A
Conclude: AB>BC
the sum of the lengths of any two sides of a triangle is greater then the length of the third side
Given: ?ABC
Conclude: AB+BC>CA, BC+CA>AB, CA+AB>BC
SAS inequality theorem
if 2 sides of 1 ? are ? to 2 sides of another ?, but the included ? of the 1st ? is > the included ? of the 2nd ?, then the 3rd side of the 1st ? is > the 3rd side of the 2nd ?
SSS inequality theorem
if 2 sides of 1 ? are ? to 2 sides of another ?, but the 3rd side of the 1st ? is > the 3rd side of the 2nd ?, the the ? opposite the 3rd side of the 1st ? is > than the ? opposite the 3rd side of the 2nd ?