Geometry Theorems, Postulates, And Definitions

Front 1

ratio

Back 1

quotient of two numbers

Front 2

proportion

Back 2

an equation stating that two or more ratios are equal

Front 3

extremes

Back 3

the first and fourth terms in a proportion

Front 4

means

Back 4

the second and third terms in a proportion

Front 5

Means-Extremes Products Theorem

Back 5

In a proportion, the product of the the means is equal to the product of the extremes.

Front 6

Means-Extremes Ratio Theorem

Back 6

If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion.

Front 7

geometric mean

Back 7

equal means in a proportion
Found by finding the square root of the product of the extremes.

Front 8

arithmetic mean

Back 8

add the two extremes together and divide the sum by two

Front 9

dilation

Back 9

enlargement by a fixed proportion

Front 10

reduction

Back 10

reduction of size by a fixed proportion

Front 11

def. similar polygons

Back 11

1. The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.
2. The ratio of any pair of corresponding sides to the ratio of another pair of corresponding sides is equal.
3. The corresponding angles of similar polygons are equal.

Front 12

AAA Postulate

Back 12

If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar.

Front 13

AA Theorem

Back 13

If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar.

Front 14

SSS~

Back 14

If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides are equal, then the triangles are similar.

Front 15

SAS~

Back 15

If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar.

Front 16

dihedral angle

Back 16

If there are 2 given noncoplanar half planes with a common edge at AB, the union of these two half planes and their common edge is called a dihedral angle.

Front 17

Altitude on Hypotenuse Theorem

Back 17

If an altitude is drawn to the hypotenuse of a right triangle, then:
a. The two triangles formed are similar to the given right triangle and to each other.
b. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse.
c. Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg.

Front 18

Pythagorean Theorem

Back 18

The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs.

Front 19

Converse of the Pythagorean Theorem

Back 19

If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.

Front 20

Distance Formula

Back 20

Line PQ = √([x2 – x1]2 + [y2 – y1]2)

Front 21

Pythagorean Triple

Back 21

Any three whole numbers that satisfy the equation (a)squared + (b)squared = (c)squared
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25

Front 22

30-60-90 Triangle Theorem

Back 22

In a triangle whose angles have the measures 30, 60, and 90, the lengths of the sides opposite these angles can be represented by x, x√3, and 2x.

Front 23

45-45-90 Triangle Theorem

Back 23

In a triangle whose angles have the measures 45, 45, and 90, the lengths of the sides opposite these angles can be represented by x, x, and x√.

Front 24

circle

Back 24

a set of all points in a plane that are a given distance from a given point in the plane

Front 25

center of a circle

Back 25

the given point of which a set of all points in a plane are a given distance away from

Front 26

radius

Back 26

the given distance that a set of all points in a plane are from a given point

Front 27

concentric circles

Back 27

two or more coplanar circles with the same center

Front 28

congruent circles

Back 28

two circles are congruent if they have congruent radii

Front 29

interior of a circle

Back 29

a point is in the interior of a circle if its distance from the center is less than the radius

Front 30

exterior of a circle

Back 30

a point is in the exterior of a circle if its distance from the center is greater than the radius

Front 31

on a circle

Back 31

a point is on a circle if its distance from the center is equal to the radius

Front 32

chord

Back 32

a segment joining any two points on the circle

Front 33

diameter

Back 33

a chord that passes through the center of the circle

Front 34

distance from the center of a circle to a chord

Back 34

the measure of the perpendicular segment from the center to the chord

Front 35

radius-chord perpendicular bisector theorem

Back 35

if a radius is perpendicular to a chord, then it bisects the chord

Front 36

converse of radius-chord perpendicular bisector theorem

Back 36

if a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord

Front 37

chord perpendicular bisector theorem

Back 37

the perpendicular bisector of a chord passese through the center of the circle

Front 38

chord equidistance theorem

Back 38

if two chords of a circle are equidistant from the center, then they are congruent

Front 39

converse of chord equidistance theorem

Back 39

if two chords of a circle are congruent, then they are equidistant from the center of the circle

Front 40

central angles-intercepted arcs theorem

Back 40

if two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent

Front 41

central angles-corresponding chords theorem

Back 41

if two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent

Front 42

arcs-corresponding chords theorem

Back 42

if two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent

Front 43

secant

Back 43

a line that intersects a circle at exactly two points

Front 44

tangent

Back 44

a line that intersects a circle at exactly one point

Front 45

point of tangency

Back 45

the point in which a tangent intersects a circle

Front 46

tangent-radius postulate

Back 46

a tangent line is perpendicular to the radius drawn to the point of contact

Front 47

converse of tangent-radius postulate

Back 47

if a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle

Front 48

tangent segment

Back 48

the part of a tangent line between the point of contact and a point outside the circle

Front 49

secant segment

Back 49

the part of a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle

Front 50

external part of a secant segment

Back 50

the part of a secant line that joins the outside point to the nearer intersection point

Front 51

two-tangent theorem

Back 51

if two tangent segments are drawn to a circle from an external point, then those segments are congruent

Front 52

tangent circles

Back 52

circles that intersect each other at exactly one point

Front 53

externally tangent circles

Back 53

two circles are externally tangent if each of the tangent circles lies outside the other

Front 54

internally tangent circles

Back 54

two circles are internally tangent if one of the tangent circles lies inside the other

Front 55

common tangent

Back 55

a line tangnet to two circles

Front 56

common internal tangent

Back 56

a common tangent that lies between the two circles

Front 57

common external tangent

Back 57

a common tangent that doesn't lie between the two circles

Front 58

inscribed angle

Back 58

an angle whose vertex is on a circle and whose sides are determined by two chords

Front 59

tangent-chord angle

Back 59

an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the tangent's point of contact

Front 60

inscribed angle theorem

Back 60

the measure of an inscribed angle or a tangent-chord angle is one-half the measure of its intercepted arc

Front 61

chord-chord angle

Back 61

an angle formed by two chords that intersect inside a circle but not at the center

Front 62

chord-chord angle theorem

Back 62

the measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle