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62 Cards in this Set
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ratio

quotient of two numbers

proportion

an equation stating that two or more ratios are equal

extremes

the first and fourth terms in a proportion

means

the second and third terms in a proportion

MeansExtremes Products Theorem

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

MeansExtremes Ratio Theorem

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.

geometric mean

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

add the two extremes together and divide the sum by two

dilation

enlargement by a fixed proportion

reduction

reduction of size by a fixed proportion

def. similar polygons

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. 
AAA Postulate

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.

AA Theorem

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.

SSS~

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.

SAS~

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.

dihedral angle

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.

Altitude on Hypotenuse Theorem

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. 
Pythagorean Theorem

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.

Converse of the Pythagorean Theorem

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.

Distance Formula

Line PQ = √([x2 – x1]2 + [y2 – y1]2)

Pythagorean Triple

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 
306090 Triangle Theorem

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.

454590 Triangle Theorem

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√.

circle

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

center of a circle

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

radius

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

concentric circles

two or more coplanar circles with the same center

congruent circles

two circles are congruent if they have congruent radii

interior of a circle

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

exterior of a circle

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

on a circle

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

chord

a segment joining any two points on the circle

diameter

a chord that passes through the center of the circle

distance from the center of a circle to a chord

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

radiuschord perpendicular bisector theorem

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

converse of radiuschord perpendicular bisector theorem

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

chord perpendicular bisector theorem

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

chord equidistance theorem

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

converse of chord equidistance theorem

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

central anglesintercepted arcs theorem

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

central anglescorresponding chords theorem

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

arcscorresponding chords theorem

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

secant

a line that intersects a circle at exactly two points

tangent

a line that intersects a circle at exactly one point

point of tangency

the point in which a tangent intersects a circle

tangentradius postulate

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

converse of tangentradius postulate

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

tangent segment

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

secant segment

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

external part of a secant segment

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

twotangent theorem

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

tangent circles

circles that intersect each other at exactly one point

externally tangent circles

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

internally tangent circles

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

common tangent

a line tangnet to two circles

common internal tangent

a common tangent that lies between the two circles

common external tangent

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

inscribed angle

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

tangentchord angle

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

inscribed angle theorem

the measure of an inscribed angle or a tangentchord angle is onehalf the measure of its intercepted arc

chordchord angle

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

chordchord angle theorem

the measure of a chordchord angle is onehalf the sum of the measures of the arcs intercepted by the chordchord angle and its vertical angle
