GEOMETRY EXAM THEOREMS

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 ?