Geometry Final Theorems

Polygon interior angles theorem

Sum of angles in a n-gon is equal to (n -2) x 180

Corollary to polygon interior angles theorem

Sum of angles in a quadrilateral equal to 360

Polygon exterior angles theorem

Sum of exterior angles in a polygon equals 360

Parallelogram opposite Angles theorem

If quadrilateral is a parallelogram → opposite angles congruent

Parallelogram opposite sides theorem

If quadrilateral is a parallelogram → opposite sides congruent

Parallelogram consecutive angles theorem

If quadrilateral is a parallelogram → consecutive angles are supplementary

Parallelogram diagonals theorem

If quadrilateral is a parallelogram → diagonals bisect each other

Parallelogram opposite angles converse

If opposite angles in a quadrilateral congruent → parallelogram

Parallelogram opposite sides converse

If opposite sides in a quadrilateral are congruent → parallelogram

Opposite sides parallel and congruent theorem

If opposite sides in a quadrilateral are parallel and congruent → parallelogram

Parallelogram diagonals converse,

If diagonals in a quadrilateral bisect each other → parallelogram

Rhombus corollary

If al 4 sides in a parallelogram are congruent → rhombus

Rectangle corollary

If all 4 angles in a parallelogram are right angles → rectangle

Square corollary

If a quadrilateral is both a rhombus and a rectangle →square

Rhombus diagonals theorem

If the diagonals of a quadrilateral are perpendicular →rhombus

Rhombus opposite angles theorem

In a quadrilateral if opposite angles are bisected by diagonals → rhombus

Rectangle diagonal theorem

Enaquadrilateral if diagonals are congruent> rectangle

Isosceles trapezoid base angles theorem

If quadrilateral is ISO, trap. → base angles congruent.

Isosceles trapezoid base angles theorem

If base angles congruent → isosceles trapezoid

Isosceles trapezoid diagonals theorem

Diagonals congruent → isosceles trapezoid

Trapezoid midsegment theorem

Midsegment = ½ ( sum of bases of trapezoid )

Kite diagonals theorem

If kite → diagonals perpendicular

Kite opposite angles theorem

If kite → one pair of opposite angles congruent

Perimeter of similar polygons

If polygons similar → ratio of perimeters = ratio of corresponding sides

Area of similar polygons

If polygons similar → ratio of areas = ( ratio of sides) ²

Angle-angle (aa) similarity theorem

Two corresponding angles congruent ( in 2 diff triangles)→ the triangles similar

Side - side-side (sss) similarity theorem

If corresponding sides are proportional →triangles similar

Side-angle-side (SAS) congruence theorem

If 2 corresponding sides proportional & the angle between is congruent →triangles similar

Triangle proportionality theorem

If a line intersects a triangle and is parallel to the base →divides intersected sides proportionally

Triangle proportionality converse

Lire splits 2 sides proportionally → line parallel to base

Three parallel lines theorem

If 3 parallel lines intersect a 2 transversals →divides transversals proportionally

Pythagorean theorem

a²+b²=c²

Pythagorean inequalities theorems

a²+b²>c² - acute

a²+b²<c² - obtuse

45 - 45-90 triangle theorem

Hypotenuse =x( sq rt 2)

Leg=x

30-60-90 triangle theorem

Leg opposite to 30 = x

Longer leg (opposite to 60) = x ( sq rt 3)

Hypotenuse e= 2 x

Right triangle similarity theorem

If altitude from hypotenuse → 2 triangles formed similar to each other and the original triangle

Law of sines

Sin (A) /a= sin ( b)/b= sin (C) /c

Law of cosines

a²=b²+c²-2bc cos(a)’

Tangent line to circle theorem

Line perpendicular to radius at its endpoint on circle → line is tangent

External tangent congruence theorem

Tangent segments from common external point congruent

Congruent circles theorem

Same radius → circles congruent

Congruent central angles theorem

Corresponding central angles congruent → minor arcs congruent

Applies to both congruent circles on within same circle

Similar circles theorem

All circles similar

Congruent corresponding chords theorem

Corresponding chords congruent → minor arcs congruent

In both congruent circles and within same circle

Perpendicular chord bisector theorem

Diameter perpendicular to chord →diameter bisects chord+ it's arc

Perpendicular chord bisector converse

one chord perpendicular to another chord → one chord diameter

Equidistance chords theorem

Chords equidistant to center → congruent

Applies to both congruent circles and within same circle

Measure of an inscribed angle theorem

Measure of inscribed angle = ½ ( measure of intercepted arc)

Inscribed angle in circle theorem

2 inscribed angles intercept same arc → angles congruent

Inscribed quadrilateral theorem

Quadrilateral inscribed in circle only if opposite angles supplementary

Tangent and intersected chord theorem

Tangent and chord intersect → measure of each angle formed = 1/2( measure of intercepted arc )

Angles inside circle theorem

2 chords intersect → measure of angle= 1/2( sum of arcs intercepted by the angle and its vertical angle)

Angles outside circle theorem

-tangent and secant

-2 Secants

2 tangents

Intersect outside a circle → measure of angle formed = 1/2(difference of measure of intercepted arcs)

Circumscribed angle theorem

Measure of circumscribed angle= 180 -( measure of central angle that intercepts same arc)

Segments of chords theorem

2 chords intersect within circle → product of lengths of segments of a chord = product of segments of other chord

Segments of secants theorem

2 secants share an external endpoint → product of external segment of secant and secant = same for the other secant

Segments of secants and tangents theorem

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