Ms.Silva's Theorems, Postulates, Definitions, and Formulas

Definition of Circumscribed about the Polygon

A circle that contains all the vertices of a polygon

Definition of a Rectangle

A quadrilateral with four right angles.

Definition of an Isosceles Trapezoid

A trapezoid in which the legs are congruent

Inscribed Quadrilateral Theorem

If a quadrilateral is inscribed in a circle, then opposite angles are supplementary

Triangle Angle-Bisector Theorem

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

Definition of Counterexample

Instance that proves the conjecture false

Reflexive POE

a = a

Rules of Rotation: Counterclockwise 180° or Clockwise 180°: (x , y)

(-x , -y)

Rules of Reflection: Reflection about the y-axis: (x , y)

(-x , y)

Rules of Rotation: Counterclockwise 90° or Clockwise 270°: (x , y)

(-y , x)

Rules of Reflection: Reflection about the line y = -x: (x , y)

(-y = -x)

Rule of Inference: Simplfication(and)

(p ^ q) → p OR (p ^ q) → q (Pick one)

Rules of Reflection: Reflection about the x-axis: (x , y)

(x , -y)

Rules of Rotation: Counterclockwise 270° or Clockwise 90°: (x , y)

(y , -x)

Rules of Reflection: Reflection about the line y = x: (x , y)

(y = x)

Definition of Sphere

A 3-dimensional object shaped like a ball where every point on the surface is the same distance from the center

Definition of Diameter

A chord that contains the center of the circle

Definition of Inscribed(Circle)

A circle is inscribed in a polygon if each side of the polygon is tangent to the circle.

Definition of Common External Tangent

A common tangent that does NOT intersect a segment joining the centers of the circles

Definition of Common Internal Tangent

A common tangent that intersects a segment joining the centers of the circles

Definition of a Ratio

A comparison between quantities

Definition of a Compound Sentence

A compound sentence can contain more than one connective. When the truth value of every simple sentence is certain within the compound being formed, we have a compound statement.

Definition of a Biconditional

A compound sentence formed by combining the two conditionals p→q and q→p under a conjunction and. The biconditional is abbreviated in the words: p if and only if q. In symbolic form, the biconditional is written: p↔q

Definition of Line/Reflectional Symmetry

A figure has reflectional symmetry if a reflection maps the figure onto itself

Definition of Rotational Symmetry

A figure has rotational symmetry if a rotation maps the figure onto itself

Definition of a Composite Figure

A figure made up of simple shapes, such as triangles, rectangles, and paralellograms

Definition of Center of Dilation

A fixed point about which all other points are transformed by a dilation

Definition of a Transformation

A general term for four specific ways to manipulate a point, line, or shape. The original shape of the object is called the original image or the preimage. The final shape and position of the object is called the image. The four main types of transformations are Rotations, Translations, Dilations, and Reflections

Definition of Tangent

A line in the plane of a circle that intersects the circle in exactly one point

Definition of Secant

A line that contains a chord

Definition of Tangent(Circle)

A line that intersects a circle at exactly 1 point

Definition of Common Tangent

A line that is tangent to each of two coplanar circles

Definition of a Perpendicular Bisector

A line/segment/ray that goes through the midpoint of a segment at a right angle

Definition of Supplementary Angles

A pair of angles whose measures have the sum of 180 degrees.

Definition of Complimentary Angles

A pair of angles whose measures have the sum of 90 degrees.

Definition of an Altitude of a Triangle

A perpendicular segment from a vertex of a triangle to the line containing the opposite side

Definition of a Polygon

A plane figure formed by coplanar segments such that: 1) each segment intersects exactly once with other segments, one at each endpoint; and 2) no two segments with a common endpoint are collinear

Definition of Convex

A polygon is convex if all the diagonals of the polygon are on the inside of the polygon

Definition of an Extended Proportion

A proportion that continues. For example, 1/2 = 2/4 = 3/6 = 4/8...

Definition of a Rhombus

A quadrilateral with 4 congruent sides

Definition of a Trapezoid

A quadrilateral with exactly one pair of opposite sides parallel

Definition of a Square

A quadrilateral with four congruent sides and angles

Definition of a Kite

A quadrilateral with two pairs of consecutive distinct congruent sides

Definition of a Vector

A quantity that has both direction and magnitude. The initial point of a vector is the starting point and the terminal point of a vector is the ending point. A vector can be denoted symbolically as <a , b>

Definition of an Angle Bisector

A ray that divides an angle into 2 congruent angles

Definition of a Midsegment

A segment that joins the midpoint of two sides of a triangle

Definition of the Median of a Triangle

A segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side

Definition of Chord

A segment whose endpoints lie on a circle

Definition of a Sequence of Rigid Motions

A sequence of reflections, rotations, and/or translations

Definition of a Theorem

A statement that can be proven

Definition of a Postulate

A statement that is accepted as true without proof

Definition of a Coordinate Proof

A style of proof that uses coordinate Geometry and Algebra

Definition of a Tangent Chord Angles

A tangent chord angle has its vertex on a circle with one side tangent to the circle at the vertex and the other side containing a chord

Definition of a Reflection

A transformation across a line (Line of Reflection) such that the line of reflection is the perpendicular bisector of each segment connecting image and preimage points.

Definition of a Translation

A transformation along a vector such that the segment joining a point and its image has the same length as the vector is parallel to the vector

Definition of a Rotation

A transformation around point P, the center of rotation, such that the following is true: -Every point and its image are the same distance from P -All angles with vertex P formed by a point and its image have the same measure. This angle measure is the angle of rotation

Definition of a Rigid Motion(Isometry)

A transformation that changes the position of a figure without changing the size or shape of the figure

Definition of Dilation

A transformation that changes the size of a polygon, but not its shape. A dilation is considered enlargement when c > 1 and considered a reduction when 0 < c < 1

Definition of an Equilangular Triangle

A triangle with at least two congruent angles

Definition of an Isosceles Triangle

A triangle with at least two congruent sides

Definition of an Equilateral Triangle

A triangle with at three congruent sides

Definition of a Truth Table

A truth table is a compact way of listing symbols to show all the possible truth values for a set of sentences

Formula: Cosine(cos) in Right Triangles

Adjacent over Hypotenuse

Definition of an Interior Angle

An angle formed by two sides of a polygon with a common vertex

Definition of an Exterior Angle

An angle that forms a linear pair with an interior angle of a polygon

Definition of Inscribed Angle

An angle whose vertex is on a circle and whose sides contain chords of the circle

Definition of Central Angle

An angle whose vertex is the center of the circle

Definition of a Remote Interior Angle

An interior angle that is not adjacent to the exterior angle

Definition of Conjecture

An unproven statement

Definition of Alternate Exterior Angles

Angles that lie on opposite sides of the transversal and outside the intersected lines

Definition of Same-Side Interior Angles

Angles that lie on the same side of the transversal and between in the intersected lines

Definition of Corresponding Angles

Angles that lie on the same side of the transversal and on same sides of the intersected lines

Definition of a Regular Polygon

Any polygon which has all equal side lengths and angle measures

Definition of Radius

Any segment that joins the center to a point of a circle

Definition of Adjacent Arcs

Arcs that have exactly one point in common

Definition of Congruent Arcs

Arcs with equal measure that lie in the same circle or congruent circles

Reflexive POC

A≅A

Definition of a Midpoint

Breaks a segment into 2 congruent parts

Definition of Concentric Circles

Circles that lie in the same plane and have the same center

Definition of Congruent Circles/Spheres

Circles/SPheres that have congruent radii

Definition of a Conjunction

Compound sentences formed by using the word and to combine two simple sentences. The symbol is ^. Thus, when p and q represent simple sentences, the conjunction p and q is written symbolically as p ^ q.

Definition of a Disjunction

Compound sentences formed by using the word or to combine two simple sentences. The symbol is v. Thus, when p and q represent simple sentences, the conjunction p or q is written symbolically as p v q.

Definition of a Conditional

Compound sentences usually formed by using the words if...then to combine two simple sentences. When p and q represent simple sentences, the conjunction if p then q is written in symbols as p→q.

Definition of Tangent Circles

Coplanar circles that are tangent to the same line at the same point

Definition of a Proportion

Equivalent ratios

Definition of an Intersection

Figures intersect if they have more than one point in common

Definition of a Segment Bisector

Geometric figure that goes through a segment's midpoint

Theorem: If a diameter bisects a chord that is not a diameter, then it is perpendicular to the chord and bisects its major and minor arcs

Hey! Unrelated Image because I couldn't find one

Transitive POC

If A ≅ B and B ≅ C, then A ≅ C

Symmetric POC

If A ≅ B, then B ≅ A

Segment Addition Postulate

If B is between A and C, then AB + BC = AC

Midpoint Theorem

If M is the midpoint of line AB, then AM = 1/2 AB and MB = 1/2 AB

Angle Addition Postulate

If S is in the interior of <RQT, then m<RQS + m<SQT = m<RQT

Transitive POE

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

Addition POE

If a = b and c = d, then a + c = b + d

Subtraction POE

If a = b and c = d, then a - c = b - d

Division POE

If a = b and c ≠ 0, then a/c = b/c

Multiplication POE

If a = b, then ac = bc

Symmetric POE

If a = b, then b = a

Substitution POE

If a = b, then either a or b may be substituted for the other in any equation or inequality

Converse of the Triangle Proportionality Theorem

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

Converse of the Tangent-Radius Theorem

If a line in the plane of a circle is perpendicular to a radius at a point on teh circle, then the line is tangent to the circle

Tangent-Radius Theorem

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency

Triangle Proportionality Theorem

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

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment

Perpendicular Bisector Theorem

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

Converse of the Alternate Exterior Angles Theorem

If a transversal intersects two lines and alternate exterior angles are congruent, then the two lines are parallel

Converse of the Alternate Interior Angles Theorem

If a transversal intersects two lines and alternate interior angles are congruent, then the two lines are parallel

Converse of the Corresponding Angles Theorem

If a transversal intersects two lines and corresponding angles are congruent, then the two lines are parallel

Converse of the Same-Side Interior Angles Postulate

If a transversal intersects two lines and same-side interior angles are supplementary, then the two lines are parallel

Converse of the Equilateral Triangle Theorem

If a triangle is equiangular, then it is equilateral

Equilateral Triangle Theorem

If a triangle is equilateral, then it is equiangular.

Angle Bisector Theorem

If ray BX is the bisector of ∠ ABC, then m∠ABX = 1/2m∠ABC and m∠XBC = 1/2m∠ABC

Hypotenuse-Leg(HL) Triangle Congruence Theorem

If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent

Side-Side-Side(SSS) Triangle Similarity Theorem

If the three sides of one triangle are proportional to the corresponding sides of another traingle, then the traingles are similar

Side-Side-Side(SSS) Triangle Congruence Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

Angle-Angle-Side(AAS) Triangle Congruence Theorem

If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent

Angle-Side-Angle(ASA) Triangle Congruence Postulate

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

Congruent Complements Theorem

If two angles are complements to the same angle or congruent angles, then the angles are congruent

Congruent Supplements Theorem

If two angles are supplements to the same angle or congruent angles, then the angles are congruent

Vertical Angles Theorem

If two angles are vertical angles, then the angles are congruent

Linear Pair Theorem

If two angles form a linear pair, then they are supplementary

Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Angle-Side Relationships in Triangles

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

Angle-Angle(AA) Triangle Similarity Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar

Corresponding Parts of Congruent Figures are Congruent(CPCFC)

If two figures are congruent, then, corresponding sides are congruent and corresponding angles are congruent. We abbreviate this as CPCFC

Corresponding Angles Theorem

If two parallel lines are cut by a transversal, then corresponding angles are congruent

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent

Same-Side Interior Angles Postulate

If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary

Side-Angle-Side(SAS) Triangle Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Side-Angle Relationships in Triangles

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

Side-Angle-Side(SAS) Triangle Similarity Theorem

If two sides of one triangle are proportional to the corresponding sides and their included angles are congruent, then the triangles are similar

Corresponding Parts of Congruent Triangles are Congruent(CPCTC)

If two triangles are congruent, then, corresponding sides are congruent and corresponding angles are congruent. We abbreviate this as CPCTC

Definition of the hypothesis/antecedent

It is an assertion or a sentence that begins an argument. The antecedent usually follows the word if

Definition of the conclusion/consequent

It is an ending or a sentence that closes an argument. The consequent usually follows the word then

Definition of the Point of Concurrency

Lines that intersect at the same point

Definition of Parallel

Lines that lie in the same plane but don't intersect are parallel

Theorem: The line through an external point and the center of a circle bisects the angle formed by the two tangents from the external point

Look at the 'y'

Definition of Alternate Interior Angles

Nonadjacent angles that lie on the opposite sides of the transversal between intersecting lines

Definition of Vertical Angles

Opposite angles formed by two intersecting lines.

Formula: Tangent(tan) in Right Triangles

Opposite over Adjacent

Formula: Sine(sin) in Right Triangles

Opposite over Hypotenuse

Definition of Collinear

Points that lie in the same line

Definition of Coplanar

Points that lie in the same plane

Definition of Similar Polygons

Polygons whose corresponding angles are congruent and whose corresponding sides are proportional

Definition of Inductive Reasoning

Relies on patterns in specific cases to form a conjecture

Definition of Concentric Spheres

Spheres that have the same center

Incenter Theorem

The angle bisectors of a triangle intersect at a point that it is equidistant from each side of the triangle.

Definition of Vertex Angle

The angle formed by the legs

Definition of the Angle of Rotational Symmetry

The angle of rotational symmetry, which is greater than 0° but less than or equal to 180°, is the smallest angle of rotation that maps a figure onto itself

Definition of Base Angles

The angles that have the base as a side

Area Addition Postulate

The area of a region is equal to the sum of the areas of its non-overlapping parts

Definition of the Circumcenter

The center of the circumcircle

Definition of the Incenter of a Triangle

The center of the incircle

Centroid Theorem

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

Definition of Legs(of a triangle)

The congruent sides of the triangle

Inscribed Angle of a Diameter Theorem

The endpoints of a diameter lie on an inscribed angle if and only if the inscribed angle is a right angle

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles

Inscribed Angle Theorem

The measure of an inscribed angle is equal to half the measure of its intercepted arc

Arc Addition Postulate

The measure of the arc formed by adjacent arcs is the sum of the measures of these two arcs.

Definition of Center

The middle of a circle or sphere(equidistant from all points on the circle/sphere)

Definition of Negation

The negation of a statement is usually formed by placing the word, not within the original, or given, statement. To show the negation of a simple statement in symbolic form, we place the symbol ~ before the letter that represents the original or given statement.

Definition of Legs of a Trapezoid

The non-parallel sides of s trapezoid

Definition of Bases of a Trapezoid

The parallel sides of the trapezoid

Circumcenter Theorem

The perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the vertices of the triangle, PA = PB = PC

Definition of the Orthocenter

The point at which all of the altitudes of a triangle intersect.

Definition of the Centroid

The point at which all of the medians of a triangle intersect. The centroid is also known as the balancing point.

Definition of Point of Tangency

The point where the tangent intersects the circle

Definition of Scale Factor

The ratio of lengths of corresponding sides in the image and preimage

Definition of a Diagonal of a Polygon

The segment joining 2 consecutive vertices of a polygon

Triangle Midsegment Theorem

The segment joining the midpoints of 2 sides of a triangle is parallel to the third side and is 1/2 the length of that side

Definition of Circle

The set of points in a plane at a given distance from a given point in that plane

Definition of a Base

The side opposite of the vertex angle

30-60-90 Triangle Theorem

The smaller(opposite 30) leg is x. The longer(opposite 60) leg is x times the square root of 3. The hypotenuse(opposite 90) is 2x.

Definition of Logic

The study of reasoning is Logic. Logic is the branch of mathematics that tells whether an argument is valid or invalid

The Triangle Sum Theorem

The sum of the interior angle measures of a triangle is 180°

Triangle Inequality Theorem

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

Polygon Angle Sum Theorem

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

Parallel Postulate

Through a point P not on line l, there is exactly one line parallel to l

Definition of Distance from a Point to the Line

To determine the distance from a point to the line, you use the length of the perpendicular segment.

Definition of a Linear Pair

Two adjacent angles that sum up to 180 degrees/ or a straight angle

Definition of Adjacent Angles

Two angles that have a shared ray(side) and vertex but have NO common interior points.

Definition of Congruent

Two figures are congruent when they are exactly the same in size and shape

Definition of Perpendicular Lines

Two lines that intersect at right angles

Definition of Arc

Two points and a continous part of the circle between the points

Definition of Deductive Reasoning

Uses facts, definitions, accepted properties, or laws of logic to form a conjecture

Definition of Geometric Mean

When a,b, and x are positive numbers and a/x = x/b, then x is the geometric mean between them

Definition of a Tautology

When the last column of the truth table is all true

Definition of Connectives

Words or Phrases that allow us to form compound statements that contain two or more thoughts. These new statements will be either true or false; examples: and, or, if ... then and if and only if

Rule of Inference: Dysjunctive Syllogism(or)

[(p v q) ^ ~ p] → q OR [(p v q) ^ ~ q] → p (Opposite one means that I have same as the other)

Rule of Inference: Modus Tollens

[(p → q) ^ ~ q] → ~p (Opposite of the second implies opposite of the first)

Rule of Inference: Modus Ponens

[(p→q) ^ p] → q (Same as the first implies same as the second)

45-45-90 Triangle Theorem

in a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg

Rule of Inference: DeMorgan's Rule

~(p ^ q)↔(~ p v ~ q) OR ~(p v q)↔(~ p ^ ~ q) (Distribution but make sure to change the connective)