Definitions, Postulates and Theorems
Name:
Definitions
Name 
Definition 
Visual Clue 
Complementary 
Two angles whose measures 

Angles 
have a sum of 90o 




Supplementary 
Two angles whose measures 

Angles 
have a sum of 180o 




Theorem 
A statement that can be proven 




Vertical Angles 
Two angles formed by intersecting lines and 


facing in the opposite direction 

Transversal 
A line that intersects two lines in the same 


plane at different points 




Corresponding 
Pairs of angles formed by two lines and a 

angles 
transversal that make an F pattern 

Pairs of angles formed by two lines and a 


angles 
transversal that make a C pattern 

Alternate interior 
Pairs of angles formed by two lines and a 

angles 
transversal that make a Z pattern 

Congruent triangles 
Triangles in which corresponding parts (sides 


and angles) are equal in measure 




Similar triangles 
Triangles in which corresponding angles are 


equal in measure and corresponding sides are 


in proportion (ratios equal) 

Angle bisector 
A ray that begins at the vertex of an angle and 


divides the angle into two angles of equal 


measure 

Segment bisector 
A ray, line or segment that divides a segment 


into two parts of equal measure 

Legs of an 
The sides of equal measure in an isosceles 

isosceles triangle 
triangle 




Base of an 
The third side of an isosceles triangle 

isosceles triangle 


Equiangular 
Having angles that are all equal in measure 




Perpendicular 
A line that bisects a segment and is 

bisector 
perpendicular to it 

Altitude 
A segment from a vertex of a triangle 


perpendicular to the line containing the 


opposite side 

Page 1 of 11
Definitions, Postulates and Theorems
Definitions
Name 

Definition 



Visual Clue 


Geometric mean 

The value of x in proportion 







a/x = x/b where a, b, and x are positive 





numbers (x is the geometric mean between a 





and b) 






Sine, sin 

For an acute angle of a right triangle, the ratio 





of the side opposite the angle to the measure 





of the hypotenuse. (opp/hyp) 





Cosine, cos 

For an acute angle of a right triangle the ratio 





of the side adjacent to the angle to the measure 





of the hypotenuse. (adj/hyp) 





Tangent, tan 

For an acute angle of a right triangle, the ratio 





of the side opposite to the angle to the measure 





of the side adjacent (opp/adj) 














Algebra Postulates 








Name 

Definition 



Visual Clue 


Addition Prop. Of 

If the same number is added to equal 




equality 

numbers, then the sums are equal 



Subtraction Prop. Of 

If the same number is subtracted from equal 



equality 

numbers, then the differences are equal 



Multiplication Prop. 

If equal numbers are multiplied by the same 



Of equality 

number, then the products are equal 










Division Prop. Of 

If equal numbers are divided by the same 



equality 

number, then the quotients are equal 



Reflexive Prop. Of 

A number is equal to itself 





equality 

















Symmetric Property 

If a = b then b = a 





of Equality 








Substitution Prop. Of 

If values are equal, then one value may be 



equality 

substituted for the other. 













Transitive Property of 
If a = b and b = c then a = c 





Equality 








Distributive Property 

a(b + c) = ab + ac 




















Congruence Postulates 






Name 


Definition 


Visual Clue 


Reflexive Property of Congruence 
A â‰… A 





Symmetric Property of 
If A â‰… B,then 
B â‰… A 




Congruence 








Transitive Property of Congruence 
If A â‰… B and B â‰… C then 








A â‰… C 




Page 2 of 11
Definitions, Postulates and Theorems
Angle Postulates And Theorems
Name 
Definition 
Visual Clue 

Angle Addition 
For any angle, the measure of the whole is 


postulate 
equal to the sum of the measures of its non 



overlapping parts 


Linear Pair Theorem 
If two angles form a linear pair, then they 



are supplementary. 


Congruent 
If two angles are supplements of the same 


supplements theorem 
angle, then they are congruent. 


Congruent 
If two angles are complements of the same 


complements 
angle, then they are congruent. 


theorem 





















Right Angle 
All right angles are congruent. 


Congruence 





















Theorem 





















Vertical Angles 
Vertical angles are equal in measure 


Theorem 





















Theorem 
If two congruent angles are supplementary, 



then each is a right angle. 


Angle Bisector 
If a point is on the bisector of an angle, then 


Theorem 
it is equidistant from the sides of the angle. 


Converse of the 
If a point in the interior of an angle is 


Angle Bisector 
equidistant from the sides of the angle, then 


Theorem 
it is on the bisector of the angle. 














































Lines Postulates And Theorems 


Name 

Definition 
Visual Clue 

Segment Addition 

For any segment, the measure of the whole 


postulate 

is equal to the sum of the measures of its 






Postulate 

Through any two points there is exactly 




one line 
























Postulate 

If two lines intersect, then they intersect at 




exactly one point. 


Common Segments 

Given collinear points A,B,C and D 


Theorem 

arranged as shown, if 



â‰… 



then 



A 
B 
C 
D 









â‰… 










A 

C 
B 
C 































Corresponding Angles 

If two parallel lines are intersected by a 


Postulate 

transversal, then the corresponding angles 




are equal in measure 


Converse of 

If two lines are intersected by a transversal 


Corresponding Angles 

and corresponding angles are equal in 


Postulate 

measure, then the lines are parallel 

Page 3 of 11
Definitions, Postulates and Theorems
Lines Postulates And Theorems
Name 
Definition 
Visual Clue 
Postulate 
Through a point not on a given line, there 


is one and only one line parallel to the 


given line 

Alternate Interior 
If two parallel lines are intersected by a 

Angles Theorem 
transversal, then alternate interior angles 


are equal in measure 

Alternate Exterior 
If two parallel lines are intersected by a 

Angles Theorem 
transversal, then alternate exterior angles 


are equal in measure 

If two parallel lines are intersected by a 


Angles Theorem 
transversal, then 


are supplementary. 

Converse of Alternate 
If two lines are intersected by a transversal 

Interior Angles 
and alternate interior angles are equal in 

Theorem 
measure, then the lines are parallel 

Converse of Alternate 
If two lines are intersected by a transversal 

Exterior Angles 
and alternate exterior angles are equal in 

Theorem 
measure, then the lines are parallel 

Converse of 
If two lines are intersected by a transversal 

Interior Angles 
and 

Theorem 
supplementary, then the lines are parallel 

Theorem 
If two intersecting lines form a linear pair 


of congruent angles, then the lines are 


perpendicular 

Theorem 
If two lines are perpendicular to the same 


transversal, then they are parallel 




Perpendicular 
If a transversal is perpendicular to one of 

Transversal Theorem 
two parallel lines, then it is perpendicular 


to the other one. 




Perpendicular Bisector 
If a point is on the perpendicular bisector 

Theorem 
of a segment, then it is equidistant from 


the endpoints of the segment 

Converse of the 
If a point is the same distance from both 

Perpendicular Bisector 
the endpoints of a segment, then it lies on 

Theorem 
the perpendicular bisector of the segment 

Parallel Lines Theorem 
In a coordinate plane, two nonvertical 


lines are parallel IFF they have the same 


slope. 

Perpendicular Lines 
In a coordinate plane, two nonvertical 

Theorem 
lines are perpendicular IFF the product of 


their slopes is 

If three or more parallel lines intersect two 


Proportionality 
transversals, then they divide the 

Corollary 
transversals proportionally. 

Page 4 of 11
Definitions, Postulates and Theorems
Triangle Postulates And Theorems
Name 
Definition 
Visual Clue 
If two angles of one triangle are equal in measure 


(AA) 
to two angles of another triangle, then the two 

Similarity 
triangles are similar 

Postulate 


If the three sides of one triangle are proportional to 


(SSS) 
the three corresponding sides of another triangle, 

Similarity 
then the triangles are similar. 

Theorem 


If two sides of one triangle are proportional to two 


side SAS) 
sides of another triangle and their included angles 

Similarity 
are congruent, then the triangles are similar. 

Theorem 


Third Angles 
If two angles of one triangle are congruent to two 

Theorem 
angles of another triangle, then the third pair of 


angles are congruent 

If two sides and the included angle of one triangle 


Side 
are equal in measure to the corresponding sides 

Congruence 
and angle of another triangle, then the triangles are 

Postulate 
congruent. 

SAS 


If three sides of one triangle are equal in measure 


Congruence 
to the corresponding sides of another triangle, then 

Postulate 
the triangles are congruent 

SSS 


If two angles and the included side of one triangle 


angle 
are congruent to two angles and the included side 

Congruence 
of another triangle, then the triangles are 

Postulate 
congruent. 

ASA 


Triangle Sum 
The sum of the measure of the angles of a triangle 

Theorem 
is 180o 

Corollary 
The acute angles of a right triangle are 


complementary. 

Exterior angle 
An exterior angle of a triangle is equal in measure 

theorem 
to the sum of the measures of its two remote 


interior angles. 

Triangle 
If a line parallel to a side of a triangle intersects the 

Proportionality 
other two sides, then it divides those sides 

Theorem 
proportionally. 

Converse of 
If a line divides two sides of a triangle 

Triangle 
proportionally, then it is parallel to the third side. 

Proportionality 


Theorem 


Page 5 of 11
Definitions, Postulates and Theorems
Triangle Postulates And Theorems
Name 
Definition 
Visual Clue 
Triangle Angle 
An angle bisector of a triangle divides the opposite 

Bisector 
sides into two segments whose lengths are 

Theorem 
proportional to the lengths of the other two sides. 




If two angles and a 


side 
triangle are equal in measure to the corresponding 

Congruence 
angles and side of another triangle, then the 

Theorem 
triangles are congruent. 

AAS 


Hypotenuse 
If the hypotenuse and a leg of a right triangle are 

Leg 
congruent to the hypotenuse and a leg of another 

Congruence 
right triangle, then the triangles are congruent. 

Theorem 


HL 


Isosceles 
If two sides of a triangle are equal in measure, then 

triangle 
the angles opposite those sides are equal in 

theorem 
measure 

Converse of 
If two angles of a triangle are equal in measure, 

Isosceles 
then the sides opposite those angles are equal in 

triangle 
measure 

theorem 


Corollary 
If a triangle is equilateral, then it is equiangular 




Corollary 
The measure of each angle of an equiangular 


triangle is 60o 

Corollary 
If a triangle is equiangular, then it is also 


equilateral 

Theorem 
If the altitude is drawn to the hypotenuse of a right 


triangle, then the two triangles formed are similar 


to the original triangle and to each other. 

Pythagorean 
In any right triangle, the square of the length of the 

theorem 
hypotenuse is equal to the sum of the square of the 


lengths of the legs. 

Geometric 
The length of the altitude to the hypotenuse of a 

Means 
right triangle is the geometric mean of the lengths 

Corollary a 
of the two segments of the hypotenuse. 

Geometric 
The length of a leg of a right triangle is the 

Means 
geometric mean of the lengths of the hypotenuse 

Corollary b 
and the segment of the hypotenuse adjacent to that 


leg. 

Circumcenter 
The circumcenter of a triangle is equidistant from 

Theorem 
the vertices of the triangle. 




Incenter 
The incenter of a triangle is equidistant from the 

Theorem 
sides of the triangle. 




Page 6 of 11
Definitions, Postulates and Theorems
Triangle Postulates And Theorems
Name 
Definition 




Visual Clue 

Centriod 
The centriod of a triangle is located 2/3 of the 


Theorem 
distance from each vertex to the midpoint of the 



opposite side. 


Triangle 
A midsegment of a triangle is parallel to a side of 


Midsegment 
triangle, and its length is half the length of that 


Theorem 
side. 








Theorem 
If two sides of a triangle are not congruent, then 



the larger angle is opposite the longer side. 












Theorem 
If two angles of a triangle are not congruent, then 



the longer side is opposite the larger angle. 












Triangle 
The sum of any two side lengths of a triangle is 


Inequality 
greater than the third side length. 


Theorem 










Hinge 
If two sides of one triangle are congruent to two 


Theorem 
sides of another triangle and the third sides are not 



congruent, then the longer third side is across from 



the larger included angle. 


Converse of 
If two sides of one triangle are congruent to two 


Hinge 
sides of another triangle and the third sides are not 


Theorem 
congruent, then the larger included angle is across 



from the longer third side. 


Converse of 
If the sum of the squares of the lengths of two 


the 
sides of a triangle is equal to the square of the 


Pythagorean 
length of the third side, then the triangle is a right 


Theorem 
triangle. 






Pythagorean 
In âˆ†ABC, c is the length of the longest side. If cÂ² > 


Inequalities 
aÂ² + bÂ², then âˆ†ABC is an obtuse triangle. If cÂ² < aÂ² 


Theorem 
+ bÂ², then âˆ†ABC is acute. 












In a 


Triangle 
and the length of the hypotenuse is the length of a 


Theorem 
length times the square root of 2. 












In a 


Triangle 
hypotenuse is 2 times the length of the shorter leg, 


Theorem 
and the length of the longer leg is the length of the 



shorter leg times the square root of 3. 


Law of Sines 
For any triangle ABC with side lengths a, b, and c, 




sin A 
= 
sin B 
= 
sin C 





a 

b 


c 


Law of 
For any triangle, ABC with sides a, b, and c, 


Cosines 
a2 =b2 
+c2 



c2 = a2 

+b2 

Page 7 of 11



Definitions, Postulates and Theorems 

Plane Postulates And Theorems 


Name 

Definition 
Visual Clue 

Postulate 

Through any three noncollinear points there is exactly 




one plane containing them. 







Postulate 

If two points lie in a plane, then the line containing those 




points lies in the plane 


Postulate 

If two points lie in a plane, then the line containing those 




points lies in the plane 







Polygon Postulates And Theorems 


Name 

Definition 
Visual Clue 

Polygon Angle 
The sum of the interior angle measures of a 


Sum Theorem 
convex polygon with n sides. 


Polygon Exterior 
The sum of the exterior angle measures, one 


Angle Sum 

angle at each vertex, of a convex polygon is 


Theorem 

360Ëš. 


Theorem 

If a quadrilateral is a parallelogram, then its 





opposite sides are congruent. 

Theorem 

If a quadrilateral is a parallelogram, then its 





opposite angles are congruent. 

Theorem 

If a quadrilateral is a parallelogram, then its 





consecutive angles are supplementary. 

Theorem 

If a quadrilateral is a parallelogram, then its 





diagonals bisect each other. 

Theorem 

If one pair of opposite sides of a quadrilateral are 





parallel and congruent, then the quadrilateral is a 




parallelogram. 

Theorem 

If both pairs of opposite sides of a quadrilateral 





are congruent, then the quadrilateral is a 




parallelogram. 

Theorem 

If both pairs of opposite angles are congruent, 





then the quadrilateral is a parallelogram. 

Theorem 

If an angle of a quadrilateral is supplementary to 





both of its consecutive angles, then the 




quadrilateral is a parallelogram. 

Theorem 

If the diagonals of a quadrilateral bisect each 





other, then the quadrilateral is a parallelogram. 






Theorem 

If a quadrilateral is a rectangle, then it is a 





parallelogram. 






Theorem 

If a parallelogram is a rectangle, then its 





diagonals are congruent. 






Theorem 

If a quadrilateral is a rhombus, then it is a 





parallelogram. 






Page 8 of 11
Definitions, Postulates and Theorems
Polygon Postulates And Theorems
Name 
Definition 
Visual Clue 
Theorem 
If a parallelogram is a rhombus then its 


diagonals are perpendicular. 




Theorem 
If a parallelogram is a rhombus, then each 


diagonal bisects a pair of opposite angles. 




Theorem 
If one angle of a parallelogram is a right angle, 


then the parallelogram is a rectangle. 




Theorem 
If the diagonals of a parallelogram are 


congruent, then the parallelogram is a rectangle. 




Theorem 
If one pair of consecutive sides of a 


parallelogram are congruent, then the 


parallelogram is a rhombus. 




Theorem 
If the diagonals of a parallelogram are 


perpendicular, then the parallelogram is a 


rhombus. 




Theorem 
If one diagonal of a parallelogram bisects a pair 


of opposite angles, then the parallelogram is a 


rhombus. 




Theorem 
If a quadrilateral is a kite then its diagonals are 


perpendicular. 




Theorem 
If a quadrilateral is a kite then exactly one pair 


of opposite angles are congruent. 




Theorem 
If a quadrilateral is an isosceles trapezoid, then 


each pair of base angles are congruent. 




Theorem 
If a trapezoid has one pair of congruent base 


angles, then the trapezoid is isosceles. 




Theorem 
A trapezoid is isosceles if and only if its 


diagonals are congruent. 




Trapezoid 
The midsegment of a trapezoid is parallel to 

Midsegment 
each base, and its length is one half the sum of 

Theorem 
the lengths of the bases. 

Page 9 of 11
Definitions, Postulates and Theorems
Polygon Postulates And Theorems
Name 

Definition 










Visual Clue 


Proportional 

If the similarity ratio of two similar figures is 
a 
, 



Perimeters and 







b 



Areas Theorem 

then the ratio of their perimeter is 
a 
and the 




























b 







ratio of their areas is 
a2 
a 
2 








or 










b2 









b 





Area Addition 

The area of a region is equal to the sum of the 




Postulate 

areas of its nonoverlapping parts. 



































Circle Postulates And Theorems 













Name 

Definition 










Visual Clue 


Theorem 

If a line is tangent to a circle, then it is perpendicular 





to the radius drawn to the point of tangency. 




Theorem 

If a line is perpendicular to a radius of a circle at a 






point on the circle, then the line is tangent to the 






circle. 













Theorem 

If two segments are tangent to a circle from the 






same external point then the segments are 






congruent. 













Arc Addition 

The measure of an arc formed by two adjacent arcs 




Postulate 

is the sum of the measures of the two arcs. 




Theorem 

In a circle or congruent circles: congruent central 






angles have congruent chords, congruent chords 






have congruent arcs and congruent acrs have 






congruent central angles. 













Theorem 

In a circle, if a radius (or diameter) is perpendicular 






to a chord, then it bisects the chord and its arc. 










Theorem 

In a circle, the perpendicular bisector of a chord is a 





radius (or diameter). 




















Inscribed 

The measure of an inscribed angle is half the 




Angle 

measure of its intercepted arc. 









Theorem 















Corollary 

If inscribed angles of a circle intercept the same arc 






or are subtended by the same chord or arc, then the 






angles are congruent 













Theorem 

An inscribed angle subtends a semicircle IFF the 






angle is a right angle 




















Theorem 

If a quadrilateral is inscribed in a circle, then its 






opposite angles are supplementary. 

























Page 10 of 11
Definitions, Postulates and Theorems
Circle Postulates And Theorems
Name 
Definition 


Visual Clue 


Theorem 
If a tangent and a secant (or chord) intersect on a 





circle at the point of tangency, then the measure of 





the angle formed is half the measure of its 





intercepted arc. 





Theorem 
If two secants or chords intersect in the interior of a 





circle, then the measure of each angle formed is half 





the sum of the measures of the intercepted arcs. 




Theorem 
If a tangent and a secant, two tangents or two 





secants intersect in the exterior of a circle, then the 





measure of the angle formed is half the difference of 





the measure of its intercepted arc. 




If two chords intersect in the interior of a circle, 




Product 
then the products of the lengths of the segments of 




Theorem 
the chords are equal. 





Secant 
If two secants intersect in the exterior of a circle, 




Secant 
then the product of the lengths of one secant 




Product 
segment and its external segment equals the product 




Theorem 
of the lengths of the other secant segment and its 





external segment. 





Secant 
If a secant and a tangent intersect in the exterior of a 




Tangent 
circle, then the product of the lengths of the secant 




Product 
segment and its external segment equals the length 




Theorem 
of the tangent segment squared. 




Equation of a 
The equal of a circle with center (h, k) and radius r 




Circle 
is (x â€“ h)2 + (y â€“ k)2 
= r2 











Other 






Name 

Definition 


Visual Clue 





























Page 11 of 11