Final Postulates, Theorems and Formulas

Perpendicular Bisector Theorem

In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Converse of the Perpendicular Bisector Theorem

In a plane, if a point is equidistant from the endpoints of a line segment, then the point lies on the perpendicular bisector of the segment.

Angle Bisector Theorem

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

Converse of the Angle Bisector Theorem

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

Circumcenter Theorem

The circumcenter is equidistant from the vertices of a triangle.

Incenter Theorem

The incenter of a triangle is equidistant from the sides of the triangle.

Centroid Theorem

The centroid of a triangle is 2/3 of the distance from each vertex to the midpoint of the opposite side

Circumcenter

point of concurrency of perpendicular bisectors of a triangle

incenter

point of concurrency of angle bisectors of a triangle

centroid

point of concurrence of medians in a triangle

orthocenter

point of concurrency of the altitudes of a triangle

midsegment of a triangle

a segment that connects the midpoints of two sides of a triangle

midsegment triangle

triangle formed by the midsegments of a triangle

triangle midsegment theorem

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

indirect proof

proof by contradiction which assumes the negation of the statement to show a contradiction.

steps for an indirect proof

1. assume the opposite is true

2. reason logically until you reach a conclusion

3. point out the desired conclusion to be true

triangle longer side theorem

if one triangle is longer than another side, than the angle opposite 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, then the side opposite the larger angle is longer than the side opposite the smaller angle.

triangle inequality theorem

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

hinge theorem

if two sides of one triangle are congruent to two sides of another triangle, and the included angle 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

converse of the hinge theorem

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 angle included by the two sides of the first is larger than that of the second.

polygon interior angles theorem

the sum of the measures of the interior angles of a polygon with n sides is given by the formula (n-2) × 180 degrees.

polygon exterior angles theorem

the sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 degrees

corollary to the polygon interior angles theorem

the sum of angles in a quadrilateral is 360 degrees

parallelogram

a quadrilateral with both pairs of opposite sides parallel

parallelogram opposite sides theorem

opposite sides are congruent in a parallelogram

parallelogram opposite angles theorem

the opposite angles are congruent in a parallelogram.

parallelogram consecutive angles theorem

the consecutive angles are supplementary in a parallelogram.

parallelogram diagonals theorem

the diagonals bisect each other in a parallelogram.

parallelogram opposite sides converse

if both pairs of opposite sides in a quadrilateral are congruent, then it is a parallelogram

parallelogram opposite angles converse

if both pairs of opposite angles in a quadrilateral are congruent, then it is a parallelogrampar

opposite sides parallel and congruent theorem

if the opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram

parallelogram diagonals theorem

if the diagonals bisect each other in a quadrilateral, then it is a parallelogram

rhombus corollary

a parallelogram is a rhombus if and only if it has four congruent sides

rectangle corollary

a parallelogram is a rectangle if and only if it has four right angles

square corollary

a parallelogram is a square if and only if it is a rhombus and a rectangle

rhombus diagonals theorem

a parallelogram is a rhombus if and only if the 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

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

a trapezoid is isosceles if and only if its diagonals are congruent

trapezoid midsegment theorem

a trapezoid midsegment is parallel to each base, and its length is ½ the sum of the lengths of the bases

kite

a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent

kite diagonals theorem

if a quadrilateral is a kite, then the diagonals are perpendicular

kite opposite angles theorem

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

corresponding parts of similar polygons

angle measures are preserved, and corresponding side lengths are similar

perimeters of similar polygons

if two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths

areas of similar polygons

if two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of the side lengths

AA Similarity Theorem

if 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar

SSS Similarity Theorem

if the corresponding side lengths of two triangles are proportional, then the triangles are similar

SAS Similarity Theorem

if an angle of one triangle is congruent to another angle and the corresponding included sides of both triangles are proportional, then the triangles are similar

Triangle Proportionality theorem

if a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally

triangle proportionality converse

if a line divides two sides of a triangle proportionally, then it is parallel to the other side

three parallel lines theorem

if three parallel lines intersect two transversals, then they divide the transversals proportionally

triangle angle bisector theorem

if a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides

Pythagorean inequalities theorem

if c²<a²+b², then △ABC is acute.

if c²>a²+b², then △ABC is obtuse

45-45-90 theorem

x:x:x√2

30-60-90 theorem

x:x√3:2x

right triangle similarity theorem

if an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other

geometric mean (ALTITUDE) theorem

in a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments whose geometric mean is equal to the altitude

geometric mean (LEG) theorem

the lengths of each leg of a right triangle with an altitude to the hypotenuse is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg

tangent ratio

opp/adj

sine ratio

opp/hyp

cosine ratio

adj/hyp

angle of depression

the angle that a downward line of sight makes with a horizontal line

angle of elevation

the angle that an upward line of sight makes with a horizontal line

area of a triangle

A=½ab sin C

Law of Sines

Sin A/a = Sin B/b = Sin C/c

• ASA and AAS case, ASS Case

Tangent Line to Circle Theorem

in a plane, a line is tangent to a circle if and only if the line is perpendicular to the radius of the circle

external tangent congruence theorem

tangent segments are congruent at a common external point

Arc addition postulate

the measure of an arc formed by 2 adjacent arc is the sum of the measures of the 2 arcs

Congruent circles theorem

two circles are congruent if and only if they have the same radius

congruent central angles theorem

in the same/congruent circles circles, 2 minor arcs are congruent if and only if their corresponding central angles are congruent

similar circles theorem

all circles are similar

similar arcs

arcs that have the same measure

congruent arcs

arcs with the same measure and are in the same/congruent circles

congruent corresponding chords theorem

in same/congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent

perpendicular chord bisector theorem

if a diameter of a circle is perpendicular to a chord, then it bisects the chord and arc

perpendicular chord bisector converse

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

equidistant chords theorem

in same/congruent circles, 2 chords are congruent if and only of they are equidistant from the center

subtend

opposite of intercept

measure of an inscribed angle theorem

the measure of an inscribed angle is ½ the measure of its intercepted arc

inscribed angles in a circle theorem

if 2 inscribed angles of a circle intercept the same arc, then the angles are congruent

inscribed right triangle theorem

if a right triangle is inscribed in a circle, then the hypotenuse is a diameter. converse is also true

inscribed quadrilateral theorem

a quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary

tangent and intersected chord theorem

if a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is ½ the measure of its intercepted arc

angles inside the circle theorem

if 2 chords intersect inside a circle, then the measure of each angles is ½ the sum of the measures of the arcs intercepted by the angle

angles outside the circle theorem

if two lines intersect outside of a circle, then the angle measure is ½ the difference of the measures of the intercepted arcs

circumscribed angle

an angle whose sides are tangent to a circle

circumscribed angle theorem

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

segments of chords theorem

if 2 chords intersect each other inside a circle, then the product of the lengths of the segments of one chord is equal to that of the other chord

segments of secants theorem

if 2 secant segments share the same external endpoint, then the products of the lengths of the secant segments and their external segments are equalse

segments of secants and tangents theorem

if a secant and a tangent share and external endpoint, then the product of the secant segment and its external segment is equal to the tangent segment squared

standard equation of a circle

(x-h)² + (y-k)² = r²

converting between degrees and radians

dimensional analysis with 180/pi

area of a sector

measure of arc/360 * πr²

Area of an inscribed polygon

½ans

volume of a prism

V=Bh

volume of a cylinder

V = πr²h