Geometry Final (6,7,10,11)

Incenter Definition

The point of Congruence of the three Angle bisectors of a triangle

Incenter Theorem

The ____ of a triangle is equidistant from the sides of a triangle

circumcenter definition

The point of concurrency of the three perpendicular bisectors of a triangle

circumcenter theorem

The ____ of a triangle is equidistant from the vertices of the triangle

converse of angle bisector theorem

If a point is in the interior of an angle and is equidistant from the two sides of the Angle then it lies on the bisector of the angle

Angle bisector theorem

If a point lies on the bisector of an angle then it is equidistant from the two sides of the angle

perpendicular bisector theorem

in a plane if a point lies on the perpendicular bisector of a segment than it is equidistant from the endpoints of the segment

converse of perpendicular bisector theorem

in a plane if a point is equidistant from the endpoints of a segment than it lies on the ____ of the segment

Centroid definition

The point of concurrency between three medians of a triangle

Centroid Theorem

A ____ of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side

triangle midsegment theorem

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

triangle longer side theorem

if one side of a triangle is longer than another side then the angle opposite to the longer side is larger than the angle opposite the shorter side

triangle larger Angle Theorem

if one angle of a triangle is larger than another angle then the side opposite the angle is larger than the side opposite the smaller angle

triangle inequality Theorem

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

Hinge Theorems

if two sides of one triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second then the third side of the first is longer than the third side of the second

Hinge Theorem converse

if two sides of one triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the second then the included angle of the first is larger than the included angle of the second

circumfrence of a circle

πd or 2πr

Arc length formula

mAB/360 × 2πr

Area of a circle

πr²

Area of a sector

mAB/360 x πr²

Area of rhombus

½ d1(d2)

Area of regular polygons

½ A(p)

Volume of prism

V=Bh

volume of cylinder

V=Bh or πr²(h)

volume of pyramid

V= 1/3Bh

Surface area of right cone

S=πr²+πr(l)

Volume of a cone

V=1/3Bh or 1/3πr²(h)

Surface area of sphere

S=4πr²

Volume of sphere

V=4/3πr²

Tangent line to circle theorem

Tangent line must be perpendicular to radius in order to be tangent

external tangent congruence theorem

if two tangents share the same external point then they are congruent

Arc addition postulate

if you add two adjacent arcs then the sum is equal to the measure of the larger arc.

congruent circles theorem

circles are only congruent if they share the same radius

congruent central angles theorem

two minor arcs if and only if their corresponding central angles are congruent.

similar circles theorem

all circles are similar

congruent corresponding chords theorem

two minor arcs are congruent if their corresponding chords are congruent

perpendicular chord bisector theorem

if the diameter of the circle is perpendicular to a chord then the diameter bisects the chord and its arc evenly

perpendicular chord bisector converse

if one chord is a perpendicular bisector to another chord then the first chord is the diameter

equidistant chord theorem

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

measure of an inscribed angle

½ the measure of its intercepted arc

inscribed angles of a circle

if two inscribed angles intercept the same arc they are congruent

inscribed right triangle

if an inscribed triangle is right then the hypotenuse is the diameter of the triangle

inscribed quadrilateral

opposite angles are supplementary

tangent and intersected chord theorem

measure of each angle formed is one half of the intercepted angle

angles inside the circle theorem

the angle is ½ of the sum of the two angles it intersects on both sides

angles outside circles theorem

angles made outside the circle with two tangents, two secants or a secant and a tangent can be calculated by finding half of the difference of the angles that the circumscribed angle intersects

circumscribed angle theorem

180- the measure of a central angle that intercepts the same arc is the measure of the circumscribed angle

Segments of chords theorem

if two chords intersect in the middle of a circle then the products of the individual chord segments equal each other

segments of secant theorem

If two secants intersect outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

segments of secants and tangents theorem

if a tangent and a secant meet at the same external point the the measure of the tangent squared equals the product of the secant and the external segment

Polygon interior angles theorem

the sum of the measures of the interior angles of a convex polygon are equal to (n-2)180where n is the number of sides of the polygon.

Corollary to the polygon interior angles theorem

the sum the interior angles of every quadrilateral is 360

polygon exterior angles theorem

the sum of the exterior angles are always 360

Parallelogram opposite sides theorem

if a quadrilateral is a parallelogram then the opposite sides are congruent and parallel to each other.

Parallelogram opposite angles theorem

if a quadrilateral is a parallelogram then the opposite angles are congruent.

parallelogram consecutive angles theorem

if a quadrilateral is a parallelogram, then each pair of consecutive angles is supplementary.

parallelogram diagonals theorem

if a quadrilateral is a parallelogram, then its diagonals bisect each other, meaning each segment is congruent

opposite sides parallel and congruent theorem

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

parallelogram diagonals converse

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

rhombus corollary

a quadrilateral is a rhombus if and only if all for of its sides are congruent

rectangle corollary

a quadrilateral is a rectangle if and only if it has four right angles.

square corollary

a quadrilateral is a square if and only if it has four congruent sides and four right angles.

rhombus diagonals theorem

a parallelogram is a rhombus if and only ifs its diagonals are perpendicular

rhombus opposite angles theorem

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

rectangle diagonals theorem

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

isosceles trapezoid base angles theorem

if a trapezoid is isosceles then each pair of base angles is congruent (top two and bottom two congruent)

isosceles trapezoid base angles converse

if a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

isosceles trapezoid diagonals theorem

the diagonals of an isosceles trapezoid are congruent.

trapezoid midsegment theorem

midsegment= ½ (top side+bottom side)

kite diagonals theorem

if a quadrilateral is a kite then its diagonals are perpendicular

kite opposite angles theorem

if a quadrilateral is a kite then exactly one pair of opposite angles are congruent