postulates and theorems list

8/11/2019 Postulates and Theorems List

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DEFINITIONSName Definition Visual Clue

Theorem A statement that can be proven

Complementary AnglesTwo angles whose measureshave a sum of 90 o

SupplementaryAngles

Two angles whose measureshave a sum of 180 o

Adjacent AnglesTwo coplanar angles with a common side,a common vertex, and no common interior

points

Linear PairTwo adjacent angles whose non-commonsides are opposite rays.

Vertical AnglesTwo angles formed by intersecting linesand facing in the oppositee direction

TransversalA line that intersects two lines in the same

plane at different points

Corresponding anglesPairs of angles formed by two lines and atransversal that make an F pattern

Same-side interior anglesPairs of angles formed by two lines and atransversal that make a C pattern

Alternate interior angles Pairs of angles formed by two lines and atransversal that make a Z pattern

Congruent trianglesTriangles in which corresponding parts(sides and angles) are equal in measure

Similar trianglesTriangles in which corresponding anglesare equal in measure and correspondingsides are in proportion (ratios equal)

Angle bisectorA ray that begins at the vertex of an angleand divides the angle into two angles of

equal measure

Segment bisector A ray, line or segment that divides asegment into two parts of equal measure

Legs of an isosceles triangleThe sides of equal measure in an isoscelestriangle

Base of an isosceles triangle The third side of an isosceles triangle

EquiangularHaving angles that are all equal inmeasure


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DEFINITIONSName Definition Visual Clue

Perpendicular bisectorA line that bisects a segment and is

perpendicular to it

AltitudeA segment from a vertex of a triangle

perpendicular to the line containing theopposite side

Geometric mean

The value of x in proportiona/x = x/b where a, b, and x are positivenumbers (x is the geometric mean

between a and b)

Sine, SIN

For an acute angle of a right triangle, theratio of the side opposite the angle to themeasure of the hypotenuse. (opp/hyp)

Cosine, COSFor an acute angle of a right triangle theratio of the side adjacent to the angle tothe measure of the hypotenuse. (adj/hyp)

Tangent, TANFor an acute angle of a right triangle, theratio of the side opposite to the angle tothe measure of the side adjacent (opp/adj)

CONGRUENCEName Definition Visual Clue

Reflexive Property ofCongruence A A

Symmetric Property ofCongruence

If then B A , A B

Transitive Property ofCongruence

If B A and C B then C A

PLANESName Definition Visual Clue

PostulateThrough any three noncollinear points there isexactly one plane containing them.

PostulateIf two points lie in a plane, then the line containingthose points lies in the plane

PostulateIf two points lie in a plane, then the line containingthose points lies in the plane


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LINESName Definition Visual Clue

Segment Additionpostulate

For any segment, the measure of the whole isequal to the sum of the measures of its non-overlapping parts

PostulateThrough any two points there is exactly one

line

PostulateIf two lines intersect, then they intersect atexactly one point.

Common SegmentsTheorem

Given collinear points A,B,C and D arrangedas shown, if AB CD then AC BD

CorrespondingAngles Postulate

If two parallel lines are intersected by atransversal, then the corresponding angles areequal in measure

Converse ofCorresponding

Angles Postulate

If two lines are intersected by a transversaland corresponding angles are equal inmeasure, then the lines are parallel

PostulateThrough a point not on a given line, there isone and only one line parallel to the given line

Alternate InteriorAngles Theorem

If two parallel lines are intersected by atransversal, then alternate interior angles areequal in measure

Alternate ExteriorAngles Theorem

If two parallel lines are intersected by atransversal, then alternate exterior angles areequal in measure

Same-side InteriorAngles Theorem

If two parallel lines are intersected by atransversal, then same-side interior angles aresupplementary.

Converse ofAlternate InteriorAngles Theorem

If two lines are intersected by a transversaland alternate interior angles are equal inmeasure, then the lines are parallel

Converse ofAlternate Exterior

Angles Theorem

If two lines are intersected by a transversaland alternate exterior angles are equal in

measure, then the lines are parallelConverse of Same-

side Interior AnglesTheorem

If two lines are intersected by a transversaland same-side interior angles aresupplementary, then the lines are parallel

TheoremIf two intersecting lines form a linear pair ofcongruent angles, then the lines are

perpendicular

TheoremIf two lines are perpendicular to the sametransversal, then they are parallel

A B C D


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LINESName Definition Visual Clue

PerpendicularTransversal

Theorem

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to theother one.

PerpendicularBisector Theorem

If a point is on the perpendicular bisector of asegment, then it is equidistant from theendpoints of the segment

Converse of thePerpendicular

Bisector Theorem

If a point is the same distance from both theendpoints of a segment, then it lies on the

perpendicular bisector of the segment

Parallel LinesTheorem

In a coordinate plane, two nonvertical linesare parallel IFF they have the same slope.

PerpendicularLines Theorem

In a coordinate plane, two nonvertical linesare perpendicular IFF the product of theirslopes is -1.

Two-TransversalsProportionality

Corollary

If three or more parallel lines intersect twotransversals, then they divide the transversals

proportionally.

ALGEBRAName Definition Visual Clue

Addition Prop. Of equalityIf the same number is added to equalnumbers, then the sums are equal

Subtraction Prop. OfequalityIf the same number is subtracted from equalnumbers, then the differences are equal

Multiplication Prop. Ofequality

If equal numbers are multiplied by the samenumber, then the products are equal

Division Prop. Of equalityIf equal numbers are divided by the samenumber, then the quotients are equal

Reflexive Prop. Of equality A number is equal to itself

Symmetric Property ofEquality

If a = b then b = a

Substitution Prop. Ofequality

If values are equal, then one value may besubstituted for the other.

Transitive Property ofEquality

If a = b and b = c then a = c

Distributive Property a(b + c) = ab + ac


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ANGLESName Definition Visual Clue

Angle Additionpostulate

For any angle, the measure of the whole isequal to the sum of the measures of its non-overlapping parts

Linear PairTheorem

If two angles form a linear pair, then they aresupplementary.

Congruentsupplements

theorem

If two angles are supplements of the sameangle, then they are congruent.

Congruentcomplements

theorem

If two angles are complements of the sameangle, then they are congruent.

Right Angle

CongruenceTheoremAll right angles are congruent.

Vertical AnglesTheorem

Vertical angles are equal in measure

TheoremIf two congruent angles are supplementary,then each is a right angle.

Angle BisectorTheorem

If a point is on the bisector of an angle, thenit is equidistant from the sides of the angle.

Converse of theAngle Bisector

Theorem

If a point in the interior of an angle isequidistant from the sides of the angle, thenit is on the bisector of the angle.


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TRIANGLESName Definition Visual Clue

S i m i l a r i t y

(AA) Angle-Angle Similarity

Postulate

If two angles of one triangle are equal in measure totwo angles of another triangle, then the two trianglesare similar

(SSS) Side-side-side SimilarityTheorem

If the three sides of one triangle are proportional tothe three corresponding sides of another triangle,then the triangles are similar.

(SAS) Side-angle-sideSimilarityTheorem

If two sides of one triangle are proportional to twosides of another triangle and their included anglesare congruent, then the triangles are similar.

C o n g r u e n c e

(SAS) Side-Angle-Side

CongruencePostulate

If two sides and the included angle of one triangleare equal in measure to the corresponding sides and

angle of another triangle, then the triangles arecongruent.

(SSS) Side-side-side Congruence

Postulate

If three sides of one triangle are equal in measure tothe corresponding sides of another triangle, then thetriangles are congruent

(ASA) Angle-side-angle

Congruence

Postulate

If two angles and the included side of one triangleare congruent to two angles and the included side ofanother triangle, then the triangles are congruent.

(HL) Hypotenuse-Leg

CongruenceTheorem

If the hypotenuse and a leg of a right triangle arecongruent to the hypotenuse and a leg of anotherright triangle, then the triangles are congruent.

(AAS) Angle-angle-side

CongruenceTheorem

If two angles and a non-included side of one triangleare equal in measure to the corresponding angles andside of another triangle, then the triangles arecongruent.

Third Angles TheoremIf two angles of one triangle are congruent to twoangles of another triangle, then the third pair ofangles are congruent

Triangle Sum TheoremThe sum of the measure of the angles of a triangle is180 o

CorollaryThe acute angles of a right triangle arecomplementary.


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Exterior angle theoremAn exterior angle of a triangle is equal in measure tothe sum of the measures of its two remote interiorangles.

Side-Splitter Theorem

If a line parallel to a side of a triangle intersects the

other two sides, then it divides those sides proportionally.Converse of Triangle

ProportionalityTheorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Triangle Angle BisectorTheorem

An angle bisector of a triangle divides the oppositesides into two segments whose lengths are

proportional to the lengths of the other two sides.

Isosceles triangle

theorem

If two sides of a triangle are equal in measure, then

the angles opposite those sides are equal in measure

Converse of Isoscelestriangle theorem

If two angles of a triangle are equal in measure, thenthe sides opposite those angles are equal in measure

Corollary If a triangle is equilateral, then it is equiangular

CorollaryThe measure of each angle of an equiangular

triangle is 60o

Corollary If a triangle is equiangular, then it is also equilateral

TheoremIf the altitude is drawn to the hypotenuse of a righttriangle, then the two triangles formed are similar tothe original triangle and to each other.

Pythagorean theoremIn any right triangle, the square of the length of thehypotenuse is equal to the sum of the square of thelengths of the legs.

Geometric MeansCorollary a

The length of the altitude to the hypotenuse of a

right triangle is the geometric mean of the lengths ofthe two segments of the hypotenuse.

Geometric MeansCorollary b

The length of a leg of a right triangle is thegeometric mean of the lengths of the hypotenuse andthe segment of the hypotenuse adjacent to that leg.

Circumcenter TheoremThe circumcenter of a triangle is equidistant fromthe vertices of the triangle.

Incenter TheoremThe incenter of a triangle is equidistant from thesides of the triangle.


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Centriod TheoremThe centriod of a triangle is located 2/3 of thedistance from each vertex to the midpoint of theopposite side.

Triangle MidsegmentTheorem

A midsegment of a triangle is parallel to a side oftriangle, and its length is half the length of that side.

TheoremIf two sides of a triangle are not congruent, then thelarger angle is opposite the longer side.

TheoremIf two angles of a triangle are not congruent, thenthe longer side is opposite the larger angle.

Triangle InequalityTheorem

The sum of any two side lengths of a triangle isgreater than the third side length.

Hinge Theorem

If two sides of one triangle are congruent to twosides of another triangle and the third sides are notcongruent, then the longer third side is across fromthe larger included angle.

Converse of HingeTheorem

If two sides of one triangle are congruent to twosides of another triangle and the third sides are notcongruent, then the larger included angle is acrossfrom the longer third side.

Converse of thePythagorean Theorem

If the sum of the squares of the lengths of two sidesof a triangle is equal to the square of the length ofthe third side, then the triangle is a right triangle.

PythagoreanInequalities Theorem

In ABC, c is the length of the longest side. If c >a + b, then ABC is an obtuse triangle. If c < a +

b, then ABC is acute.

45 -45 -90 TriangleTheorem

In a 45 -45 -90 triangle, both legs are congruent,and the length of the hypotenuse is the length of alength times the square root of 2.

30 -60 -90 TriangleTheorem

In a 30 -60 -90 triangle, the length of thehypotenuse is 2 times the length of the shorter leg,and the length of the longer leg is the length of theshorter leg times the square root of 3.

T R I G Law of Sines

For any triangle ABC with side lengths a, b, and c,

c

C

b

B

a

A sinsinsin

Law of CosinesFor any triangle, ABC with sides a, b, and c,

C acbac

Baccab Abccba

cos2

,cos2,cos2222

222222


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POLYGONS & QUADRILATERALSName Definition Visual Clue

Polygon AngleSum Theorem

The sum of the interior angle measures of a convex polygon with n sides is 180( n - 2)

PolygonExterior AngleSum Theorem

The sum of the exterior angle measures, one angle ateach vertex, of a convex polygon is 360.

TheoremIf a quadrilateral is a parallelogram, then its oppositesides are congruent.

TheoremIf a quadrilateral is a parallelogram, then its oppositeangles are congruent.

Theorem If a quadrilateral is a parallelogram, then itsconsecutive angles are supplementary.

TheoremIf a quadrilateral is a parallelogram, then its diagonals

bisect each other.

TheoremIf one pair of opposite sides of a quadrilateral are

parallel and congruent, then the quadrilateral is a parallelogram.

Theorem If both pairs of opposite sides of a quadrilateral arecongruent, then the quadrilateral is a parallelogram.

TheoremIf both pairs of opposite angles are congruent, thenthe quadrilateral is a parallelogram.

TheoremIf an angle of a quadrilateral is supplementary to bothof its consecutive angles, then the quadrilateral is a

parallelogram.

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

TheoremIf a quadrilateral is a rectangle, then it is a

parallelogram.

TheoremIf a parallelogram is a rectangle, then its diagonalsare congruent.


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POLYGONS & QUADRILATERALSName Definition Visual Clue

TrapezoidMidsegment

Theorem

The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengthsof the bases.

ProportionalPerimeters andAreas Theorem

If the similarity ratio of two similar figures is ba

, then

the ratio of their perimeter isb

a and the ratio of their

areas is 22

b

a or

2

b

a

Area AdditionPostulate

The area of a region is equal to the sum of the areasof its nonoverlapping parts.

CIRCLESName Definition Visual Clue

TheoremIf a line is tangent to a circle, then it is perpendicularto the radius drawn to the point of tangency.

TheoremIf a line is perpendicular to a radius of a circle at a

point on the circle, then the line is tangent to thecircle.

TheoremIf two segments are tangent to a circle from the sameexternal point then the segments are congruent.

Arc AdditionPostulate

The measure of an arc formed by two adjacent arcs isthe sum of the measures of the two arcs.

Theorem

In a circle or congruent circles: congruent centralangles have congruent chords, congruent chords havecongruent arcs and congruent acrs have congruentcentral angles.

TheoremIn a circle, if a radius (or diameter) is perpendicular toa chord, then it bisects the chord and its arc.

TheoremIn a circle, the perpendicular bisector of a chord is a

radius (or diameter).Inscribed

AngleTheorem

The measure of an inscribed angle is half the measureof its intercepted arc.

CorollaryIf inscribed angles of a circle intercept the same arcor are subtended by the same chord or arc, then theangles are congruent

TheoremAn inscribed angle subtends a semicircle IFF theangle is a right angle


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CIRCLESName Definition Visual Clue

TheoremIf a quadrilateral is inscribed in a circle, then itsopposite angles are supplementary.

Theorem

If a tangent and a secant (or chord) intersect on acircle at the point of tangency, then the measure ofthe angle formed is half the measure of its interceptedarc.

TheoremIf two secants or chords intersect in the interior of acircle, then the measure of each angle formed is halfthe sum of the measures of the intercepted arcs.

Theorem

If a tangent and a secant, two tangents or two secantsintersect in the exterior of a circle, then the measureof the angle formed is half the difference of themeasure of its intercepted arc.

Chord-ChordProductTheorem

If two chords intersect in the interior of a circle, thenthe products of the lengths of the segments of thechords are equal.

Secant-Secant

ProductTheorem

If two secants intersect in the exterior of a circle, thenthe product of the lengths of one secant segment andits external segment equals the product of the lengthsof the other secant segment and its external segment.

Secant-TangentProductTheorem

If a secant and a tangent intersect in the exterior of acircle, then the product of the lengths of the secant

segment and its external segment equals the length ofthe tangent segment squared.

Equation of aCircle

The equal of a circle with center (h, k) and radius r is(x h)2 + (y k) 2 = r 2

OtherName Definition Visual Clue

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