Postulates And Theorems

Front 1

Segment Addition Postulate

Back 1

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

Front 2

Reflexive Property

Back 2

A quantity is congruent (equal) to itself. a = a

Front 3

Symmetric Property

Back 3

If a = b, then b = a.

Front 4

Transitive Property

Back 4

If a = b and b = c, then a = c.

Front 5

Addition Postulate

Back 5

If equal quantities are added to equal quantities, the sums are equal.

Front 6

Subtraction Postulate

Back 6

If equal quantities are subtracted from equal quantities, the differences are equal.

Front 7

Multiplication Postulate

Back 7

If equal quantities are multiplied by equal quantities, the products are equal.

Front 8

Division Postulate

Back 8

If equal quantities are divided by equal nonzero quantities, the quotients are equal.

Front 9

Substitution Postulate

Back 9

A quantity may be substituted for its equal in any expression.

Front 10

Partition Postulate

Back 10

The whole is equal to the sum of its parts.
Also: Between of Points: AB + BC = AC
Angle Addition Postulate: m<ABC + m<CBD = m<ABD

Front 11

Construction

Back 11

Two points determine a straight line.

Front 12

Construction

Back 12

From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Front 13

Right Angles

Back 13

All right angles are congruent.

Front 14

Straight Angles

Back 14

All straight angles are congruent.

Front 15

Congruent Supplements

Back 15

Supplements of the same angle, or congruent angles, are congruent.

Front 16

Congruent Complements

Back 16

Complements of the same angle, or congruent angles, are congruent.

Front 17

Linear Pair

Back 17

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

Front 18

Vertical Angles

Back 18

Vertical angles are congruent.

Front 19

Triangle Sum

Back 19

The sum of the interior angles of a triangle is 180º.

Front 20

Exterior Angle

Back 20

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle.

Front 21

Base Angle Theorem

Back 21

If two sides of a triangle are congruent, the angles opposite these sides are congruent.

Front 22

Base Angle Converse

Back 22

If two angles of a triangle are congruent, the sides opposite these angles are congruent.

Front 23

Side-Side-Side (SSS) Congruence

Back 23

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Front 24

Side-Angle-Side (SAS) Congruence

Back 24

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Front 25

Angle-Side-Angle (ASA) Congruence

Back 25

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Front 26

Angle-Angle-Side (AAS) Congruence

Back 26

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Front 27

Hypotenuse-Leg (HL) Congruence (right triangle)

Back 27

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.

Front 28

Theorem

Back 28

Corresponding parts of congruent triangles are congruent.

Front 29

Angle-Angle (AA) Similarity

Back 29

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Front 30

Side-Side-Side (SSS) Similarity

Back 30

If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.

Front 31

Side-Angle-Side (SAS) Similarity

Back 31

If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.

Front 32

Side Proportionality

Back 32

If two triangles are similar, the corresponding sides are in proportion.

Front 33

Mid-segment Theorem

Back 33

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.

Front 34

Sum of Two Sides

Back 34

The sum of the lengths of any two sides of a triangle must be greater than the third side

Front 35

Longest Side

Back 35

In a triangle, the longest side is across from the largest angle.
In a triangle, the largest angle is across from the longest side.

Front 36

Altitude Rule

Back 36

The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.

Front 37

Leg Rule

Back 37

Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

Front 38

Corresponding Angles Theorem

Back 38

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Front 39

Corresponding Angles Converse

Back 39

If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.

Front 40

Alternate Interior Angles Postulate

Back 40

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

Front 41

Alternate Exterior Angles Postulate

Back 41

If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

Front 42

Interiors on Same Side

Back 42

If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.

Front 43

Alternate Interior Angles
Converse

Back 43

If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.

Front 44

Alternate Exterior Angles
Converse

Back 44

If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel.

Front 45

Interiors on Same Side Converse

Back 45

If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.

Front 46

Parallelograms

Back 46

If a quadrilateral is a parallelogram, the opposite sides are parallel.

Front 47

Parallelograms

Back 47

If a quadrilateral is a parallelogram, the opposite sides are congruent.

Front 48

Parallelograms

Back 48

If a quadrilateral is a parallelogram, the opposite angles are congruent.

Front 49

Parallelograms

Back 49

If a quadrilateral is a parallelogram, the consecutive angles are supplementary.

Front 50

Parallelograms

Back 50

If a quadrilateral is a parallelogram, the diagonals bisect each other.

Front 51

Parallelograms

Back 51

If a quadrilateral is a parallelogram, the diagonals form two congruent triangles.

Front 52

Parallelogram Converses

Back 52

If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram.

Front 53

Parallelogram Converses

Back 53

If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

Front 54

Parallelogram Converses

Back 54

If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.

Front 55

Parallelogram Converses

Back 55

If the consecutive angles of a quadrilateral are supplementary, the quadrilateral is a parallelogram.

Front 56

Parallelogram Converses

Back 56

If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

Front 57

Parallelogram Converses

Back 57

If the diagonals of a quadrilateral form two congruent triangles, the quadrilateral is a
parallelogram.

Front 58

Parallelogram

Back 58

If one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.

Front 59

Rectangle

Back 59

If a parallelogram has one right angle it is a rectangle.

Front 60

Rectangle

Back 60

A parallelogram is a rectangle if and only if its diagonals are congruent.

Front 61

Rectangle

Back 61

A rectangle is a parallelogram with four right angles.

Front 62

Rhombus

Back 62

A rhombus is a parallelogram with four congruent sides.

Front 63

Rhombus

Back 63

If a parallelogram has two consecutive sides congruent, it is a rhombus.

Front 64

Rhombus

Back 64

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

Front 65

Rhombus

Back 65

A parallelogram is a rhombus if and only if the diagonals are perpendicular.

Front 66

Square

Back 66

A square is a parallelogram with four congruent sides and four right angles.

Front 67

Square

Back 67

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Front 68

Trapezoid

Back 68

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Front 69

Isosceles Trapezoid

Back 69

An isosceles trapezoid is a trapezoid with congruent legs.

Front 70

Isosceles Trapezoid

Back 70

A trapezoid is isosceles if and only if the base angles are congruent

Front 71

Isosceles Trapezoid

Back 71

A trapezoid is isosceles if and only if the diagonals are congruent

Front 72

Isosceles Trapezoid

Back 72

If a trapezoid is isosceles, the opposite angles are supplementary.

Front 73

Radius

Back 73

In a circle, a radius perpendicular to a chord bisects the chord and the arc.

Front 74

Radius

Back 74

In a circle, a radius that bisects a chord is perpendicular to the chord.

Front 75

Radius

Back 75

In a circle, the perpendicular bisector of a chord passes through the center of the circle.

Front 76

Radius

Back 76

If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.

Front 77

Chords

Back 77

In a circle, or congruent circles, congruent chords are equidistant from the center. (and converse)

Front 78

Chords

Back 78

In a circle, or congruent circles, congruent chords have congruent arcs. (and converse)

Front 79

Chords

Back 79

In a circle, parallel chords intercept congruent arcs

Front 80

Chords

Back 80

In the same circle, or congruent circles, congruent central angles have congruent chords (and converse)

Front 81

Tangents

Back 81

Tangent segments to a circle from the same external point are congruent

Front 82

Arcs

Back 82

In the same circle, or congruent circles, congruent central angles have congruent arcs. (and converse)

Front 83

Angles

Back 83

An angle inscribed in a semi-circle is a right angle.

Front 84

Angles

Back 84

In a circle, inscribed angles that intercept the same arc are congruent.

Front 85

Angles

Back 85

The opposite angles in a cyclic quadrilateral are supplementary

Front 86

Angles

Back 86

In a circle, or congruent circles, congruent central angles have congruent arcs.