Theorem

Front 1

2-1 Vertical Angles Theorem

Back 1

Vertical angles are congruent.

Front 2

2-2 Congruent Supplements Theorem

Back 2

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Front 3

2-3 Congruent Complements Theorem

Back 3

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Front 4

2-4

Back 4

All right angles are congruent.

Front 5

2-5

Back 5

If two angles are congruent and supplementary, then each is a right angle.

Front 6

3-1 Alternate Interior Angles Theorem

Back 6

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

Front 7

3-2 Corresponding Angles Theorem

Back 7

If a transversal intersects two parallel lines, then corresponding angles are congruent.

Front 8

3-3 Alternate Exterior Angles Theorem

Back 8

If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

Front 9

3-4 Converse of the Corresponding Angles Theorem

Back 9

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

Front 10

3-5 Converse of the Alternate Interior Angles Theorem

Back 10

If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.

Front 11

3-6 Converse of the Same-Side Interior Angles Postulate

Back 11

If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.

Front 12

3-7 Converse of the Alternate Exterior Angles Theorem

Back 12

If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel.

Front 13

3-8

Back 13

If two lines are parallel to the same line, then they are parallel to each other.

Front 14

3-9

Back 14

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

Front 15

3-10 Perpendicular Transversal Theorem

Back 15

In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

Front 16

3-11 Triangle Angle Sum Theorem

Back 16

The sum of the measures of the angles of a triangle is 180

Front 17

3-12 Triangle Exterior Angle Theorem

Back 17

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles

Front 18

4-1 Third Angles Theorem

Back 18

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Front 19

4-2 Angle-Angle-Side (AAS)

Back 19

If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent.

Front 20

4-3 Isosceles Triangle Theorem

Back 20

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

Front 21

4-4 Converse of the Isosceles Triangle Theorem

Back 21

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

Front 22

4-5

Back 22

If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.

Front 23

Corollary to Theorem 4-3

Back 23

If a triangle is equilateral, then the triangle is equiangular.

Front 24

Corollary to Theorem 4-4

Back 24

If a triangle is equiangular, then the triangle is equilateral.

Front 25

4-6 Hypotenuse-Leg (HL) Theorem

Back 25

If the hypotenuse and a leg of one right triangle is congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Front 26

5-1 Triangle Midsegment Theorem

Back 26

If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long.

Front 27

5-2 Perpendicular Bisector Theorem

Back 27

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment.

Front 28

5-3 Converse of the Perpendicular Bisector Theorem

Back 28

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Front 29

5-4 Angle Bisector Theorem

Back 29

If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

Front 30

5-5 Converse of the Angle Bisector Theorem

Back 30

If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

Front 31

5-6 Concurrency of Perpendicular Bisectors Theorem

Back 31

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Front 32

5-7 Concurrency of Angle Bisectors Theorem

Back 32

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

Front 33

5-8 Concurrency of Medians Theorem

Back 33

The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.

Front 34

5-9 Concurrency of Altitudes Theorem

Back 34

The lines that contain the altitudes of a triangle are concurrent.

Front 35

Corollary to the Triangle Exterior Angle Theorem

Back 35

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

Front 36

5-10

Back 36

If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

Front 37

5-11

Back 37

If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

Front 38

5-12 Triangle Inequality Theorem

Back 38

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Front 39

5-13 The Hinge Theorem (SAS Inequality Theorem)

Back 39

If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. (If m mYZ)

Front 40

5-14 Converse of the Hinge Theorem (SSS) Inequality Theorem

Back 40

If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side. (If BC>YZ, then m m

Front 41

6-1 Polygon Angle-Sum Theorem

Back 41

The sum of the interior measures of the angles of an n-gon is (n-2)180.

Front 42

Corollary to the Polygon Angle-Sum Theorem

Back 42

The measure of each interior angle of a regular n-gon is [(n-2)180]/n.

Front 43

6-2 Polygon Exterior Angle-Sum Theorem

Back 43

The sum of the measure of the exterior angles of a polygon, one at each vertex, is 360.

Front 44

6-3

Back 44

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Front 45

6-4

Back 45

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Front 46

6-5

Back 46

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Front 47

6-6

Back 47

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Front 48

6-7

Back 48

If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Front 49

6-8

Back 49

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

Front 50

6-9

Back 50

If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

Front 51

6-10

Back 51

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

Front 52

6-11

Back 52

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

Front 53

6-12

Back 53

If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

Front 54

6-13

Back 54

If a parallelogram is a rhombus, then it's diagonals are perpendicular.

Front 55

6-14

Back 55

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Front 56

6-15

Back 56

If a parallelogram is a rhombus, then it's diagonals are congruent.

Front 57

6-16

Back 57

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Front 58

6-17

Back 58

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Front 59

6-18

Back 59

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Front 60

6-19

Back 60

If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.

Front 61

6-20

Back 61

If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Front 62

6-21

Back 62

If a quadrilateral is a trapezoid, then

(1) the midsegment is parallel to the bases

(2) the length of the midsegment is half the sum of the lengths of the bases.

Front 63

6-22

Back 63

If a quadrilateral is a kite, then its diagonals are perpendicular.

Front 64

7-1 Side-Angle-Side Similarity (SAS~) Theorem

Back 64

If an angle of one triangle is congruent to an angle of another triangle, and the sides that include the two angles are proportional, then the triangles are similar.

Front 65

7-2 Side-Side-Side Similarity (SSS~) Theorem

Back 65

If the corresponding sides of two triangles are proportional, then the triangles are similar.

Front 66

7-3

Back 66

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

Front 67

Corollary 1 to 7-3

Back 67

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

Front 68

Corollary 2 to 7-3

Back 68

The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment to the hypotenuse adjacent to the leg.

Front 69

7-4 Side-Splitter Theorem

Back 69

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Front 70

Corollary to the Side-Splitter Theorem

Back 70

If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

Front 71

7-5 Triangle-Angle-Bisector Theorem

Back 71

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Front 72

8-1 Pythagorean Theorem

Back 72

If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Front 73

8-2 Converse of the Pythagorean Theorem

Back 73

If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

Front 74

8-3

Back 74

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.

Front 75

8-4

Back 75

If the square of the length of the longest side of a triangle is lesser than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

Front 76

8-5 45-45-90 Triangle Theorem

Back 76

In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is the square root of 2 time the length of a leg.

Front 77

8-6 30-60-90 Triangle Theorem

Back 77

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is root 3 times the length of the shorter leg.

Front 78

9-1

Back 78

The composition of two or more isometries is an isometry.

Front 79

9-2 Reflections Across Parallel Lines

Back 79

A composition of reflections across two parallel lines is a translation.

Front 80

9-3 Reflections Across Intersecting Lines

Back 80

A composition across two intersecting lines is a rotation.

Front 81

10-6 Area of a Regular Polygon

Back 81

The area of a regular polygon is half the product of the apothem and the perimeter.

A=1/2 ap

Front 82

10-11 Area of a Circle

Back 82

The area of a circle is the product of pi and the square of the radius.

Front 83

10-1 Area of a Rectangle

Back 83

The area of a rectangle is the product of its base and height

Front 84

10-2 Area of a Parallelogram

Back 84

The area of a parallelogram is the product of a base and the corresponding height

Front 85

10-3 Area of a Triangle

Back 85

The area of a triangle is half the product of a base and the corresponding height.

Front 86

10-4 Area of a Trapezoid

Back 86

The area of a trapezoid is half the product of the height and the sum of the bases.

Front 87

10-5 Area of a Rhombus or a Kite

Back 87

The area of a rhombus or a kite is half the product of the length of its diagonals

Front 88

10-8 Area of a Triangle Given SAS

Back 88

The area of a triangle is half the product of the lengths of two sides and the sine of the included angle.

Front 89

11-1 Lateral and Surface Areas of a Prism

Back 89

The lateral area of a right prism is the product of the perimeter of the base and the height of the prism.

L.A.=ph

The surface area of a right prism is the sum of the lateral area and the areas of the two bases.

S.A.=L.A. + 2B

Front 90

11-2 Lateral and Surface Areas of a Cylinder

Back 90

The lateral area of a cylinder is the product of he circumference of the base and the height of the cylinder.

L.A.=2πr*h or L.A.=πdh

The surface area of a cylinder is the sum of the lateral area and the areas of the two bases.

S.A.=L.A.+2B or S.A.=2πrh+2πr(squared)

Front 91

11-3 Lateral and Surface Areas of a Pyramid

Back 91

The lateral area of a regular pyramid is half the product of the perimeter, p, of the base and the slant height, l, of the pyramid.

L.A.=1/2pl

The surface area of a regular pyramid is the sum of the lateral area and the area, B, of the base.

S.A.=L.A.+B

Front 92

11-4 Lateral and Surface Areas of a Cone

Back 92

The lateral area of a right cone is half the product of the circumference of the base and the slant height of the cone.

L.A.=1/2(2πr)(l) or L.A.=πrl

The surface area of a cone is the sum of the lateral area and the area of the base.

S.A.=L.A.+B

Front 93

10-8 Area of a Triangle Given SAS

Back 93

The area of the triangle is half the product of the lengths of two sides and the sine of the included angle.

Front 94

12-4

Back 94

Theorem: Within a circle or in congruent circles, congruent central angles have congruent arcs.

Converse: Within a circle or in congruent circles, congruent arcs have congruent central angles.

Front 95

12-5

Back 95

Theorem: Within a circle or in congruent circles, congruent central angles have congruent chords.

Converse: Within a circle or in congruent circles, congruent chords have congruent central angles.

Front 96

12-6

Back 96

Theorem: Within a circle or in congruent circles, congruent chords have congruent arcs.

Converse: Within a circle or in congruent circles, congruent arcs have congruent chords.

Front 97

12-7

Back 97

Theorem: Within a circle or in congruent circles, chords equidistant from the center or centers are congruent.

Converse: Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers).

Front 98

12-8

Back 98

In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

Front 99

12-9

Back 99

In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord.

Front 100

12-10

Back 100

In a circle, the perpendicular bisector of a chord contains the center of the circle

Front 101

10-12 Area of a Sector of a Circle

Back 101

The area of a sector of a circle is the produt of the ratio (measure of the arc/360) and the area of the circle.

Front 102

12-1

Back 102

If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.

Front 103

12-2

Back 103

If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then he line is tangent to the circle.

Front 104

12-3

Back 104

If two tangent segments to a circle share a comment endpoint outside the circle, then the two segments are congruent.

Front 105

12-11 Inscribed Angle Theorem

Back 105

The measure of an inscribed angle is half the measure of its intercepted arc.

Front 106

Corollarys to 12-11

Back 106

1: Two inscribed angles that intercept the same are congruent

2: An angle inscribed on a semicircle is a right angle

3: The ooposite angles of a quaadrilateral inscribed in a circle are supplementary

Front 107

12-12

Back 107

Th emasure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Front 108

12-13

Back 108

The measure of a angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs.

Front 109

12-14

Back 109

The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.

Front 110

12-15

Back 110

For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle.

Front 111

12-16 Equation of a Circle

Back 111

An equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2=r^2