external secant segmant- the part of a secent segmant that is out side of the circle.

Secant segament-the part of the secant segmantthat extends form an extends from an an exterior from an exterior point the\at includes a chord of the circle.

Tangent segment- the point ofa tangent that extends from an exterior from an exterior

Theorem 10-15: if two chord intersect instead a circle,then the product of the

## Thursday, April 17, 2008

April 9, 2008

Glossary Terms

-Arc

-Arc Length

-Arc Measure

-Center of a circle

-Central angle of a circle

-Chord

-Diameter

-Intercepted Arc

-Major Arc

-Minor Arc

-Radius

-Semicircle

Theorems, Postulates, and Definitions

-Circle: is the set of all points in a plane that are equidistant from the given point in the plane known as the center of a circle.

-Radius: a segment that goes from the center to the circle.

-All radii in the same or congruent circles are congruent

-Chord: segment within a circle that intersects the circle in 2 points.

-Congruent chords are equidistant from the center

-Diameter: the longest chord of a circle and contains the center of the circle.

Radius Chord Diameter

Arcs

-A chord divides a circle into 2 arcs.

Minor Arc Major Arc

-A third point must be added to the circle in order to identify a major arc

-Central Angle: the central angle of a circle is an angle in the plane of a circle whose vertex is the center of the circle, or more simply, the central angle is the angle between 2 radii.

B

A

Arc Measurement

-Arcs are measured in 2 ways:

The degree measure of the arc: this is the measure of an angle that the arc would intercept

The length of the arc: the actual part of the circumference that makes up the arc

Degree Measure of Arcs:

-The degree measure of a minor arc is the measure of its central angle.

-The degree measure of a major arc is 360° minus the degree measure of its central angle.

-The degree measure of a semicircle is 180°

Theorems, Postulates, &Definitions

Arc Length: If r is the radius of a circle and M is the degree measure of an arc of the circle, then the length, L, of the arc is given by:

L= M/360 (2πr)

April 14th, 2008

Inscribed angles – and inscribed angle is an angle made up of 2 chords. Since it is made up of chords, the vertex of the angle will be ON the circle.

There are 3 types of inscribed angles, Center Interior, Center Exterior, Center included.

Intercepted Arc – the minor arc defined by the 2 endpoints of chords forming an inscribed angle that are not part of the vertex of the inscribed angle.

Theorems, Postulates, and Definitions

-Inscribed Angle Theorem 10 -5: the measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc (or, the measure of an intercepted arc is twice the measure of the inscribed angle.)

-Theorem 10-6: Arc-Intercepted Theorem: if 2 inscribed angles intercept congruent arcs or the same arc, the angles are congruent.

-Theorem 10-7: Right-Angle Theorem: if an inscribed angle intercepts a semicircle, then the angle is a right angle.

Inscribed Quadrilateral Theorem

-Theorem 10-8: If a quadrilateral is inscribed in a circle then it’s opposite angles are supplementary.

April 16th, 2008

Secants, Tangents, and Angle measures

-Secant-a line that intersects a circle in 2 points

-segments AB and CB are secants.

Interior Angle Theorem

-Angles that are formed by 2 intersecting chords.

D

A B

C

-Interior Angle Theorem: the measure of the angle formed by the 2 intersecting chords is equal to ½ the sum of the measures of the intercepted arcs.

Exterior Angles

-an angle formed by 2 secants, 2 tangents, or a secant and a tangent drawn from a point outside the circle.

-Exterior Angle Theorem-the measure of the angle formed is equal to ½ the difference of the intercepted arcs.

## Monday, April 14, 2008

Intercepted Arc- the minor arc defined by the two endpoints of chords forming an inscribed angle that are not part of the vertex of the inscribed angle.

INscribed Angle Theorem 10-5: the measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc (or, the measure of an intercepted arc arc is twice the measure of the inscribed angle.

The arc intercept theorem: if two inscribed angles intercept congruent arcs or the same arc, the angles are congruent

the right angle theorem -if an inscribed angle intercepts a simicircle, then the angle is a right angle.

Inscribed qaudrillateral theorem- if a qudrilateralis inscribed ina circle then its oppisite angles are supplemetary

## Wednesday, April 9, 2008

arc

arc length

arc measure

center of a circle

central angle of a circle

chord

diameter

intercepted arc

major arc

minor arc

radius

semicircle

Defintions

Cicle the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.

Radius: a segment that goes from the center and goes to the circle.

Chord: a segment in a circle that intersects the circle in two points

Diameter the longest chord af a circle and it contains the center of the cicle.

arc: chord divides the circle into two arcs.

Central Angle: a central angle of a circle is an angle of a circle whose vertex is the center of the circle

## Tuesday, April 1, 2008

Theorem 8-17: If the diagonals of a parallelogram bisect the opposite angles of the parallelogram then the parallelogram is a rhombusSummary Rhombi1) A rhombus has all the properties of a parallelogram.2) All sides are congruent3) Diagonals are perpendicular4)Diagonals bisect the angles of the rhombusSummary Squares1) A square has all the properties of a parallelogram2) A square has all of the properties of the rectangle3) A square has all of the properties of the rhombusTrapezoidWhat is a trapezoid?A trapezoid is a quadrilateral with exactly one set of parallel sidesTrapezoid Bases: The parallel sides of the trapezoid.Trapezoid Legs: The non-parallel sides of a trapezoidIsosceles Trapezoid: If the legz of a trapezoid are congruent then the trapezoid is isoceles.Median of a trapezoid- Is the segment that connects the medpoints of the legs of a trapezoid.Theorems, Postulates, & Definitions

· Theorem 8-18: Both pairs of base angles of an isosceles trapezoid are congruent

· Theorem 8-19: The diagonals of an isosceles trapezoid are congruent.

Math Notes March 6, 2008Polygon Angle Sum TheoremsObjectives:· To Classify Polygons· To find the sum of the measures of the interior and exterior angles.Polygon:· A closed plane figure.· With at least 3 sides (segments)· The sides only intersect at their endpoints· Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.Example one:II. Also Classify polygons by their shapea) Convex Polygon: Has no diagonalWith points outside the polygon.b) Concave Polygon: Has at leastOne diagonal with points outside the polygon.a)

Math Notes March 11, 2007Using a calculator will give you an approximation.

· Math Notes;· Check to see if it is a rectangle as well as a rhombus:· W(1,10); X(-4,0); Y(7,2); Z(12,12)· The diagonals are not congruent, so this is not a rectangle or a square.

· Math Notes;· Check to see if it is a rectangle as well as a rhombus:· W(1,10); X(-4,0); Y(7,2); Z(12,12)· The diagonals are not congruent, so this is not a rectangle or a square

RootsSquare RootsWhen working a square root problem. Ask: -- what times itself is the number inside the root symbol?”=3 because 3 times 3 is 9because 5 times 5 is 25Roots and Prime Numbers=3 because 3x3x3 is 27. The small 3 outside the root symbol tells how many times the answer must be multiplied to get the number inside the root.because 2x2x2x2x2 or 25=32Prime Number: a number with only factor one and itself 2,3,5,7,9,11,13,1,7,19,23… are prime numbers fifteen is not prime because 3 and 5 also divide it evenly. 15 is a composite number.Prime factorization.Prime Factorization is writing a number using multiplication of only prime numbers.12 can be written as 3x4 but 4 is not prime and can be written as 2x2So the prime factorization of 12 is 3x2x2 this can also be written 3x22Graphing Prime Numbers33010 x 335 x2 3 x 11To write 330 using its prime factorization start breaking it down by each number.Simplifying RootsYou won’t be using the radical button on your calculator anymore.= 2Prime Factorization

Definition- RectangleA rectangle is a parallelogram with four right angles.PROPERTIES1. Opposite sides are congruent and parallel.2. Opposite angles are congruent.3. Consecutive angles are supplementary.4. Diagonals bisect each other and are congruent.5. All four angles are right angles.Theorems, Postulates, & definitions.Theorem 8-13: If a parallelogram is a rectangles the diagonals are congruent.Note: If one angle of a parallelogram is a right angle then the parallelogram is a rectangle.The Housebuilder Theorem 8-14: If the diagonals of a parallelogram are congruent then the parallelogram are congruent then the parallelogram is a rectangle.Rhombi and SquaresTheorem 8-15: The diagonals of a rhombus is perpendicular.Theorem 8-16: If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

Theorems, Postulates, and Definitions.Theorem 8-8: If two pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 8-10: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 8-11: If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.Theorem 8-12: If one pair of opposite sides of a quadrilateral are parellel and congruent, then the quadrilateral is a parrelellagram.

Summary!

A quadrilateral is a parallelogram if any one of the following statements are true.1. Both pairrs of opposite sides are parrallel. (definition)2. Both pairs of opposite sides are congruent. (theorem 8-9)3. Both pairs of opposite angles are congruent (throrem -10)4. The diagnols bisect each other (theorem 8-11)5. A pair of opposite sides is both parellel and congruent (theorem 8-12)

Parallelograms in the coordinate plane

The slope formula can be used to determine if the opposite sides have the same slope.

The distance formula can be used to see if the opposite sides are congruent or

The slope and the distance formula ccan be used to determine if one pair of opposite sides is parallel and congruent.

· Theorem 8-18: Both pairs of base angles of an isosceles trapezoid are congruent

· Theorem 8-19: The diagonals of an isosceles trapezoid are congruent.

Math Notes March 6, 2008Polygon Angle Sum TheoremsObjectives:· To Classify Polygons· To find the sum of the measures of the interior and exterior angles.Polygon:· A closed plane figure.· With at least 3 sides (segments)· The sides only intersect at their endpoints· Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.Example one:II. Also Classify polygons by their shapea) Convex Polygon: Has no diagonalWith points outside the polygon.b) Concave Polygon: Has at leastOne diagonal with points outside the polygon.a)

Math Notes March 11, 2007Using a calculator will give you an approximation.

· Math Notes;· Check to see if it is a rectangle as well as a rhombus:· W(1,10); X(-4,0); Y(7,2); Z(12,12)· The diagonals are not congruent, so this is not a rectangle or a square.

· Math Notes;· Check to see if it is a rectangle as well as a rhombus:· W(1,10); X(-4,0); Y(7,2); Z(12,12)· The diagonals are not congruent, so this is not a rectangle or a square

RootsSquare RootsWhen working a square root problem. Ask: -- what times itself is the number inside the root symbol?”=3 because 3 times 3 is 9because 5 times 5 is 25Roots and Prime Numbers=3 because 3x3x3 is 27. The small 3 outside the root symbol tells how many times the answer must be multiplied to get the number inside the root.because 2x2x2x2x2 or 25=32Prime Number: a number with only factor one and itself 2,3,5,7,9,11,13,1,7,19,23… are prime numbers fifteen is not prime because 3 and 5 also divide it evenly. 15 is a composite number.Prime factorization.Prime Factorization is writing a number using multiplication of only prime numbers.12 can be written as 3x4 but 4 is not prime and can be written as 2x2So the prime factorization of 12 is 3x2x2 this can also be written 3x22Graphing Prime Numbers33010 x 335 x2 3 x 11To write 330 using its prime factorization start breaking it down by each number.Simplifying RootsYou won’t be using the radical button on your calculator anymore.= 2Prime Factorization

Definition- RectangleA rectangle is a parallelogram with four right angles.PROPERTIES1. Opposite sides are congruent and parallel.2. Opposite angles are congruent.3. Consecutive angles are supplementary.4. Diagonals bisect each other and are congruent.5. All four angles are right angles.Theorems, Postulates, & definitions.Theorem 8-13: If a parallelogram is a rectangles the diagonals are congruent.Note: If one angle of a parallelogram is a right angle then the parallelogram is a rectangle.The Housebuilder Theorem 8-14: If the diagonals of a parallelogram are congruent then the parallelogram are congruent then the parallelogram is a rectangle.Rhombi and SquaresTheorem 8-15: The diagonals of a rhombus is perpendicular.Theorem 8-16: If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

Theorems, Postulates, and Definitions.Theorem 8-8: If two pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 8-10: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 8-11: If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.Theorem 8-12: If one pair of opposite sides of a quadrilateral are parellel and congruent, then the quadrilateral is a parrelellagram.

Summary!

A quadrilateral is a parallelogram if any one of the following statements are true.1. Both pairrs of opposite sides are parrallel. (definition)2. Both pairs of opposite sides are congruent. (theorem 8-9)3. Both pairs of opposite angles are congruent (throrem -10)4. The diagnols bisect each other (theorem 8-11)5. A pair of opposite sides is both parellel and congruent (theorem 8-12)

Parallelograms in the coordinate plane

The slope formula can be used to determine if the opposite sides have the same slope.

The distance formula can be used to see if the opposite sides are congruent or

The slope and the distance formula ccan be used to determine if one pair of opposite sides is parallel and congruent.

Theorem 8-17: If the diagonals of a parallelogram bisect the opposite angles of the parallelogram then the parallelogram is a rhombusSummary Rhombi1) A rhombus has all the properties of a parallelogram.2) All sides are congruent3) Diagonals are perpendicular4)Diagonals bisect the angles of the rhombusSummary Squares1) A square has all the properties of a parallelogram2) A square has all of the properties of the rectangle3) A square has all of the properties of the rhombusTrapezoidWhat is a trapezoid?A trapezoid is a quadrilateral with exactly one set of parallel sidesTrapezoid Bases: The parallel sides of the trapezoid.Trapezoid Legs: The non-parallel sides of a trapezoidIsosceles Trapezoid: If the legz of a trapezoid are congruent then the trapezoid is isoceles.Median of a trapezoid- Is the segment that connects the medpoints of the legs of a trapezoid.Theorems, Postulates, & Definitions

· Theorem 8-18: Both pairs of base angles of an isosceles trapezoid are congruent

· Theorem 8-19: The diagonals of an isosceles trapezoid are congruent.

pages434-43512-34

Math Notes March 6, 2008Polygon Angle Sum TheoremsObjectives:· To Classify Polygons· To find the sum of the measures of the interior and exterior angles.Polygon:· A closed plane figure.· With at least 3 sides (segments)· The sides only intersect at their endpoints· Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.Example one:II. Also Classify polygons by their shapea) Convex Polygon: Has no diagonalWith points outside the polygon.b) Concave Polygon: Has at leastOne diagonal with points outside the polygon.a)

Math Notes March 11, 2007Using a calculator will give you an approximation.

· Math Notes;· Check to see if it is a rectangle as well as a rhombus:· W(1,10); X(-4,0); Y(7,2); Z(12,12)· The diagonals are not congruent, so this is not a rectangle or a square.

· Math Notes;· Check to see if it is a rectangle as well as a rhombus:· W(1,10); X(-4,0); Y(7,2); Z(12,12)· The diagonals are not congruent, so this is not a rectangle or a square

Geom Notes For SQUARE ROOTS! ;]

RootsSquare RootsWhen working a square root problem. Ask: -- what times itself is the number inside the root symbol?”=3 because 3 times 3 is 9because 5 times 5 is 25Roots and Prime Numbers=3 because 3x3x3 is 27. The small 3 outside the root symbol tells how many times the answer must be multiplied to get the number inside the root.because 2x2x2x2x2 or 25=32Prime Number: a number with only factor one and itself 2,3,5,7,9,11,13,1,7,19,23… are prime numbers fifteen is not prime because 3 and 5 also divide it evenly. 15 is a composite number.Prime factorization.Prime Factorization is writing a number using multiplication of only prime numbers.12 can be written as 3x4 but 4 is not prime and can be written as 2x2So the prime factorization of 12 is 3x2x2 this can also be written 3x22Graphing Prime Numbers33010 x 335 x2 3 x 11To write 330 using its prime factorization start breaking it down by each number.Simplifying RootsYou won’t be using the radical button on your calculator anymore.= 2Prime Factorization

Definition- RectangleA rectangle is a parallelogram with four right angles.PROPERTIES1. Opposite sides are congruent and parallel.2. Opposite angles are congruent.3. Consecutive angles are supplementary.4. Diagonals bisect each other and are congruent.5. All four angles are right angles.Theorems, Postulates, & definitions.Theorem 8-13: If a parallelogram is a rectangles the diagonals are congruent.Note: If one angle of a parallelogram is a right angle then the parallelogram is a rectangle.The Housebuilder Theorem 8-14: If the diagonals of a parallelogram are congruent then the parallelogram are congruent then the parallelogram is a rectangle.Rhombi and SquaresTheorem 8-15: The diagonals of a rhombus is perpendicular.Theorem 8-16: If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

Theorems, Postulates, and Definitions.Theorem 8-8: If two pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 8-10: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 8-11: If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.Theorem 8-12: If one pair of opposite sides of a quadrilateral are parellel and congruent, then the quadrilateral is a parrelellagram.

Summary!

A quadrilateral is a parallelogram if any one of the following statements are true.1. Both pairrs of opposite sides are parrallel. (definition)2. Both pairs of opposite sides are congruent. (theorem 8-9)3. Both pairs of opposite angles are congruent (throrem -10)4. The diagnols bisect each other (theorem 8-11)5. A pair of opposite sides is both parellel and congruent (theorem 8-12)

Parallelograms in the coordinate plane

The slope formula can be used to determine if the opposite sides have the same slope.

The distance formula can be used to see if the opposite sides are congruent or

The slope and the distance formula ccan be used to determine if one pair of opposite sides is parallel and congruent.

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