Geometry Theorems And Corollaries

Front 1

Theorem 1

Back 1

Two triangles are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other (SAS)

Front 2

Corollary 1-1

Back 2

Two right triangles are congruent if the two legs of the other (L.L.)

Front 3

Corresponding parts of congruent triangles are equal

Back 3

C.P.C.T.E.

Front 4

Theorem 2

Back 4

Two triangles are congruent if a side and the two adjacent angles of one are equal respectively to a side and the two adjacent angles of the other (A.S.A.)

Front 5

Corollary 2-1

Back 5

Two right triangles are congruent if a leg and the adjacent acute angle of one are equal respectively to a leg and the adjacent acute angle of the other (L.A)

Front 6

Theorem 3

Back 6

In any isosceles triangle the angles opposite the equal sides are equal.

Front 7

Corollary 3-1

Back 7

An equilateral triangle is also equiangular

Front 8

Theorem 4

Back 8

Two triangles are congruent if the three sides of one are equal respectively to the three sides of the other (SSS)

Front 9

Theorem 5

Back 9

An exterior angle of a triangle is greater then either opposite interior angle.

Front 10

Coorollary 5-1

Back 10

One and only one perpendicular can be drawn to a line from a point outside the line

Front 11

Theorem 6

Back 11

If two angles of a triangle are equal, the sides opposite those angles are equal

Front 12

Corollary 6-1

Back 12

An equiangular triangle is also equilateral

Front 13

Theorem 7

Back 13

When two straight lines are cut by a transversal if a pair of alternate interior angles are equal the two straight lines are parallel

Front 14

Corollary 7-1

Back 14

When two straight lines are cut by a transversal if two corresponding angles are equal the two straight lines are parallel

Front 15

Corollary 7-2

Back 15

Two lines perpendicular to the same straight \line are parallel

Front 16

corollary 7-3

Back 16

When two straight lines are cut by a transversal if the sum of the two interior angles on the same side of the transversal is equal to a straight angle the two straight lines are parallel

Front 17

Theorem 8

Back 17

If two parallel lines are cut by a transversal the alternate interior angles are equal

Front 18

Corollary 8-1

Back 18

If two parallel lines are cut by a transversal then the corresponding angles are equal

Front 19

Corollary 8-2

Back 19

If two parallel lines are cut by a transversal then the sum of the two interior angles on the same side of the transversal is equal to a straight angle.

Front 20

Corollary 8-3

Back 20

When two straight lines are cut by a transversal if the sum of the interior angles on the same side of the transversal does not equal a straight angle then the two lines are not parallel

Front 21

Corollary 8-4

Back 21

A straight line perpendicular to one of two parallel lines is perpendicular to the other also

Front 22

Corollary 8-5

Back 22

Lines perpendicular to nonparallel lines are not parallel

Front 23

Theorem 9

Back 23

If two angles have their sides parallel right side to right side and lift lide to left side the angles are equal

Front 24

Corollary 9-1

Back 24

If two angles have their sides parallel right side to left side and left side to right side the angles are supplementary

Front 25

Theorem 10

Back 25

If two angles have their sides perpendicular right side to right side and left side to left side the angles are equal

Front 26

Corollary 10-1

Back 26

If two anles have their sides perpendicular right side to left side and left side to right side the angles are supplementary

Front 27

Theorem 11

Back 27

The sum of the angles of a triangle is a straight angle

Front 28

Corollary 11-1

Back 28

An exterior angle of a triangle is equal to the sum of the two opposite interior angles

Front 29

corollary 11-2

Back 29

In any triangle there can be but one right angle or obtuse angle.

Front 30

Corollary 11-3

Back 30

In any right triangle the two acute angles are complementary

Front 31

Corollary 11-4

Back 31

If an acute angle of one right triangle equals an acute angle of another right triangle the remaining acute angles of the two triangles are equal

Front 32

Corollary 11-5

Back 32

Two right triangles are congruent if the hypotenuse and an acute angle of one are equal respectively to the hypotenuse and an acute angle of the other (H.A.)

Front 33

Corollary 11-6

Back 33

Two right triangles are congruent if a leg and either acute angle of one are equal respectively to a leg and the corresponding acute angle of the other (L.A.)

Front 34

Corollary 11-7

Back 34

If two angles of one triangle are equal respectively to two angles of another then the third angle of the first is equal to the third angle of the second

Front 35

Corollary 11-8

Back 35

Two triangles are congruent if a side and any two angles of one are equal respecively to a corresponding side and two angles of the other (S.A.A.)

Front 36

Corrollary 11-9

Back 36

Each angle of an equilateral triangle is 60 degrees

Front 37

Theorem 12

Back 37

Two right triangles are congruent if the hypotenuse and a leg of one are equal respectively to the hypotenuse and a leg of the other (H.L.)

Front 38

Corollary 12-1

Back 38

The perpendicular from the vertex to the base off an isosceles triangle visects the base and also the vertex angle

Front 39

Theorem 13

Back 39

If one side of a triangle is greater than a second side the opposite the first side is greater than the angle opposite the second side

Front 40

Theorem 14

Back 40

If one angle of a triangle is greater than a second angle the side opposite the first angle is greater than the side opposite the second angle.

Front 41

Corollary 14-1

Back 41

The perpendicular is the shortest line that can ve drawn from a given point to a given line

Front 42

Corollary 14-2

Back 42

If a line is the shortest line that can be drawn from a given point to a given line then it is a perpendicular from the point to the given line

Front 43

Theorem 15

Back 43

If two straight lines drawn from a point in a perpendicular to a given line cut off equal segments from the foot of the perpendicular then they are equal

Front 44

Corollary 15-1

Back 44

If two straight lines drawn from a point in a perpendicular to a given line cut off unequal segments from the foot of the perpendicular then the straight line that cuts off the greater segment is the greater straight line

Front 45

Theorem 16

Back 45

If two straight lines drawn from a point in a perpendicular to a given line are equal then they cut off equal segments from the foot of the perpendicular

Front 46

Corollary 16-1

Back 46

If two straight lines drawn from a point in a perpendicular to a given line are unequal then the freater cuts off the greater sefments from the foot of the perpendicular

Front 47

Theorem 17

Back 47

If two triangles have two sides of one equal respectively to two sides of the other and the included angles are unequal then the triangle which has the greater included angle has the greater third side

Front 48

Theorem 18

Back 48

If two triangles have two sides of one equal respectively to two sides of the other and the third sides are unequal then the triangle which has the greater third side has the greater angle opposite the third side

Front 49

Theorem 19

Back 49

The opposite sides of a parallelogram are equal and the opposite angles are equal

Front 50

Corollary 19-1

Back 50

All the sides of a rhombus are equal and all the sides of a square are equal

Front 51

Corollary 19-2

Back 51

A diagonal divides a parallelogram into two congruent triangles

Front 52

Corollary 19-3

Back 52

Parallel lines included between parallel lines are equal

Front 53

Corollary 19-4

Back 53

`Two parallel lines are everywhere the same distance apart

Front 54

Theorem 20

Back 54

The diaonals of a parallelogram bisect each other

Front 55

Theorem 21

Back 55

If the opposite sides of a quuadrilateral are equal then the figure is a parallelogram

Front 56

Theorem 22

Back 56

If two sides of a quadrilateral are equal and parallel then the figure is a parallelogram

Front 57

Theorem 23

Back 57

If the diagonals of a quadrilateral bisect each other then the figure is a parallelogram

Front 58

Theorem 24

Back 58

If three or more parallels cut off equal segments on one transversal they cut off equal segments on any other transversal

Front 59

Corollary 24-1

Back 59

The line parallel to one side of a triangle and bisecting a second side biseccts the third side

Front 60

Theorem 25

Back 60

The line that joins the midpoints of two sides of a triangle is parallel to the third side and equals half the third side

Front 61

Theorem 26

Back 61

The sum of the angles of a polygon of n sides is (n-2) straight angles

Front 62

Corollary 26-1

Back 62

In an equiangular polygon of n sides each angle equals [(n-2) divided by n] straight angles

Front 63

Corollary 26-2

Back 63

The sum of the exteriior angles of a polygon made by extending each of its sides in succession is equal to two straight angles

Front 64

Theorem 27

Back 64

(a) Ebery point in the perpendicular bisector of a line is equidistant from the ends of that line
(b) Every point equidistant from the ends of a line lies in the perpendicular bisector of that line

Front 65

Corollary 27-1

Back 65

Two points each equidistant from the ends of a line determine the perpendicular bisector of that line

Front 66

Theorem 28

Back 66

(a) Every point in the bisector of an angle is equidistant from the sides of that angle
(b) Every point eequidistant from the sides of an angle lies in the bisector of that angle

Front 67

Theorem 29

Back 67

The perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the three vertices

Front 68

Corollary 29-1

Back 68

The three altitudes of a triangle meet in a point

Front 69

Theorem 30

Back 69

The visectors of the angles of a triangle meet in a point which is equidistant from the three sides

Front 70

Theorem 31

Back 70

The medians of a triangle meet in a point which is two thirds of distance from each vertex to the midpoint of the opposite side