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)