Chapter 6

Describing Polygons

A polygon is a figure that is:-formed by 3 or more segments called sides, such that no 2 sides with a common endpoint are collinear- each side intersects exactly 2 other sides, one at each endpoint. Each endpoint of the side is called a vertex.

P

T S

R

QVertex

Vertex

Side

Naming Polygons

# of Sides

Type of Polygon

3 triangle

4 quadrilateral

5 pentagon

6 hexagon

7 heptagon

# of Sides

Type of Polygon

8 octagon

9 nonagon

10 decagon

12 dodecagon

n n-gon

Identifying Convex and Concave

A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon.

A polygon is that is not convex is called nonconvex or concave.

Definitions:A polygon is equilateral if all of its

sides are congruent.A polygon is equiangular if all of its

interior angles are congruent.A polygon is regular if it is both

equilateral and equiangular.

Theorem 6.1: Polygon Angle-Sum TheoremThe sum of the measures of the interior

angles of an n-gon is (n-2)180.

If you draw a diagonal in a polygon, you create triangles. Using the Triangle Sum Theorem you can conclude that the sum of the measures of the interior angles of a quadrilateral is 2(180)=360°.

Corollary to Polygon Angle-Sum Theorem

The measure of each of the interior angles of a regular polygon is

(Where n is the number of sides.)

Theorem 6.2 Polygon Exterior Angle-Sum TheoremThe sum of the measures of the exterior angles

of a polygon, one at each vertex, is 360 degrees.

( 2)180n

n

PARALLELOGRAM:

A quadrilateral with both pairs of opposite sides parallel.

Theorem 6.3:

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Theorem 6.4:

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Theorem 6.5:

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Theorem 6.6:

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Special ParallelogramsA rhombus is a

parallelogram with 4 congruent sides.

A rectangle is a parallelogram with 4 right angles

A square is a parallelogram

with 4 right angles and 4

congruent sides

Rectangle RhombusSquare

Diagonals of Special Parallelograms

Thm. 6.13: If a parallelogram is a rhombus, then its diagonals are perpendicular.

Thm. 6.14: If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Thm. 6.15: If a parallelogram is a rectangle, then its diagonals are congruent.

Quadrilateral Family

Properties of TrapezoidsA trapezoid is a quadrilateral with

exactly one pair of parallel sides.

Trapezoid Terminology:The parallel sides are called BASES.   The nonparallel sides are called LEGS.  There are two pairs of base angles, the two touching the top base, and the two touching the bottom base.

ISOSCELES TRAPEZOID If the legs of a trapezoid are congruent, then the trapezoid is an

isosceles trapezoid.

Thm. 6.19: If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent. Thm. 6.20: If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Midsegment of a TrapezoidThe midsegment of a trapezoid connects

the midpoints of its legs.

Thm. 6.21: If a quadrilateral is a trapezoid, then…

1) The midsegment is parallel to both bases and

2) The length of the midsegment is half the sum of the lengths of the bases.

Kites

A kite is a quadrilateral that has 2 pairs of consecutive congruent sides, but opposite sides are not congruent.

Thm. 6.22: If a quadrilateral is a kite, then its diagonals are perpendicular.

Interesting fact: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Relationships Among Quadrilaterals

Fill in Chart

Formulas and the Coordinate Plane

FormulaDistance Formula

Midpoint Formula

Slope Formula

2

12

2

12 yyxxd

2,

22121 yyxx

When to Use it….To determine whether…• Sides are congruent• Diagonals are congruentTo determine …• The coordinates of the midpoint

of a side• Whether diagonals bisect each

otherTo determine whether…• Opposite sides are parallel• Diagonals are perpendicular• Sides are perpendicular

12

12

xx

yym

Chapter 7

Reminders on Ratios: It is a comparison of two quantities by

division

Notation: or , read a to b.

Measured in same units. Denominator can not be zero. Usually expressed in simplified form:

6:8 simplified to 3:4

a

b:a b

Proportion

a c

b d

An equation that equals two ratios.

Means

Extremes

Similar Polygons

Two polygons are SIMILAR if and only if:

1-their corresponding angles are congruent,

2- the measures of their corresponding sides are proportional.

Similar Polygons

Symbol to indicate similarity: ~ABCD ~ GHIJ

(ratio of the lengths of two corresponding sides)

Angle-Angle Similarity PostulatePostulate 7.1 Angle-Angle Similarity

(AA~) Postulate:If two angles of one triangle are congruent to

2 angles of another triangle, then the two triangles are similar.

  Hint: From earlier information we know that if we have two congruent angles then we also know the third angles are congruent.  Thus AA is the same as AAA.  This is the most common proof of two triangles to be similar.

THEOREM S

THEOREM 7.1 Side-Angle-Side (SAS~) Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

then XYZ ~ MNP.

ZXPM

XYMN

If X M and =

X

Z Y

M

P N

THEOREM S

THEOREM 7.2 Side-Side-Side (SSS~) Similarity Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

If = =A BPQ

BCQR

CARP

then ABC ~ PQR.

A

B C

P

Q R

Theorem 7.3The altitude to the hypotenuse of a

right triangle divides the triangle into 2 triangles that are similar to the original triangle and to each other.

ΔCBD ~ ΔABC

ΔACD ~ ΔABC

ΔCBD ~ ΔACD

The GEOMETRIC MEAN between two positive numbers a and b is the positive number x where   bax b

x

x

a

What is the geometric mean of 5 and 12?

What is the geometric mean of 6 and 16?

Theorem 7.4: Side-Splitter Theorem

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

A B

C

D

E BE BD

EC DA

Theorem 7.5 Triangle-Angle Bisector TheoremIf a ray bisects an angle of a triangle,

then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Chapter 8

The Pythagorean Theorem

*Remember the Pythagorean Theorem is only true for RIGHT triangles!!

*The hypotenuse (c) is always the longest side and opposite the right angle!

*The legs (a and b) are the two sides that form the right angle.

Pythagorean Triple

Classifying Triangles:

2 2 2c a b

Theorem 8.3:

If , then the triangle is obtuse.

Theorem 8.4:

If , then the triangle is acute.

2 2 2c a b

Certain triangles possess "special" properties that allow us to use "short cut formulas" in arriving at information about their measures.  These formulas let us arrive at the answer very quickly.

Theorem 8-5 - 45º-45º-90º

: : 2n n n

Theorem 8.6

: 3 : 2n n n

Hypotenuse = 2 x shorter leg

Longer Leg = x shorter leg

3

30 60 90

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

Trigonometric Ratios

Writing Trigonometric Ratios

*What are the ratios for angle G?

Angle of Elevation/Depression: The angle formed by a horizontal line and the line of sight to an object either above or below the horizontal line.

Vectors:

Vector: any quantity with both magnitude (size) and direction

The magnitude corresponds to the distance from the initial point to the terminal point of the vector

The direction corresponds to which way the arrow is pointed

Describing a Vector You can indicate a vector by using

an ordered pair. For example, <-2, 4> is a vector with its initial point at the origin and its terminal point at (-2, 4).

We use brackets to represent a vector (called Component Form)

Direction of a Vector: You can use a compass

arrangement on the coordinate grid to describe a vector’s direction.

This vector is 30 south of east.This vector is 40 east of north.

Magnitude of a Vector The magnitude of a vector is its length.

You can use the distance formula to determine the length, or magnitude, of a vector.

Adding Two Vectors

The sum of two vectors is called the Resultant

Chapter 9

TransformationThe change in the position, shape, or

size of a geometric figure.

Preimage – the original figure before the transformation; resulting figure is the image

IsometryA transformation in which the

preimage and image are congruent.

Moves ALL points in the plane the same direction and the same distance. 

Translation:

Translation (slide)A transformation that maps all points

of a figure the same distance in the same direction. A translation is an isometry.

Reflections (flip)Reflection across a line r, called the line

of reflection, is a transformation with these two properties: If a point A is on line r, then the image of A

is itself (that is A’=A) If a point B is not on line r, then r is the

perpendicular bisector of BB’

(line of reflection)

Rotations

A rotation is a transformation in which a figure is turned about a fixed point.

The fixed point is the center of rotation (rotocenter).

Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.

A rotation about a point P (rotocenter) through x degrees (x º ).

Rotations can be clockwise or counterclockwise. So, you should state the direction. Unless otherwise stated, our book uses counterclockwise rotations.

Clockwise rotation of 60°

Counterclockwise rotation of 40°

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180º or less. For instance, a square has rotational symmetry because it maps onto itself by a rotation of 90º.

0º rotation 45º rotation 90º rotation

Line Symmetry or Reflectional Symmetry

A figure which reflects upon its own image. The line of reflection is called the line of symmetry.

Rotational Symmetry A figure which rotates 180° or less back upon

itself. The angle of rotation is the smallest angle needed for the figure to rotate onto itself.

Point Symmetry A figure with 180° rotational symmetry also has

point symmetry. Each segment joining a point and its 180° rotation image passes through the center of rotation.

Dilations:A transformation that produces an image

that is the same shape as the original (pre-image), but is a different size.

Pre-Image and image(΄) are not congruent-THEY ARE SIMILAR!

Every description should include a scale factor and center of dilation

Reduction if 0<scale factor (n)<1Enlargement if scale factor (n)>1To find the scale factor (n), compare, as a

ratio, the length on the image to the length on the preimage.

• •C C

P

Q

R

P

Q

R

P´

Q´

R´

P´

Q´

R´

3

6

2

5

Reduction: n = = =36

12

CPCP

Enlargement: n = =52

CPCP

Center: C

Identifying Dilations: finding scale factor

Chapter 10

Chapter 11

An edge of a polyhedron is a segment formed by the intersection of two faces.

A vertex of a polyhedron is a point where three or more edges meet.

A polyhedron is a space figure, or three dimensional solid, whose surfaces are polygons, called faces, that enclose a single region of space.

The intersection of the plane and the solid is called a cross section.

Imagine a plane slicing through a solid.

Notice that the sum of the number of faces and vertices is two more than the number of edges in the solids in the last slide. This result was proved by the Swiss mathematician Leonhard Euler (1707 - 1783).Key Concept

Euler’s FormulaThe sum of the number of faces (F)

and vertices (V) of a polyhedron is two more than the number of its edges (E).

F + V = E + 2

PrismsPrism: a polyhedron with 2

congruent faces, or bases, that lie in parallel planes.

The other faces are called lateral faces.

The segments connecting these faces are lateral edges.

Prisms are classified by their bases.

base

base

Lateral face

Lateral edge

The altitude (height) of a prism is the perpendicular distance between its bases.

In a right prism, each lateral edge is perpendicular to both bases.

In an oblique prism, the lateral edges are not perpendicular to the bases.

oblique

right

altitude

altitude

Lateral and Surface AreaLateral Area: the sum of the areas of

the lateral faces.Surface Area: the sum of the area of the

faces

Theorem 11.1: Lateral and Surface Areas of a Right Prism:

LA = PhSA = LA + 2B or Ph + 2B where:

B is the area of the base P is the perimeter of base h is the height of prism

B

B

P h

DefinitionsA cylinder is a solid with congruent

circular bases that lie in parallel planes.A cylinder is called a right cylinder if the

segment joining the centers of the bases is perpendicular to the bases.

In a cylinder that is oblique, the segment joining the centers is perpendicular to the planes containing the bases.

The lateral area of a cylinder is the area of its curved surface, which is

actually a rectangle.

Oblique Cylinder Right Cylinder

Theorem 11.2: Lateral and Surface Area of a Cylinder:

LA = 2πr · h or πdh SA = LA + 2B or 2πrh + 2πr2

where: B is the area of the base

h is the height of cylinderr is the radius of the base

Finding the Surface Area of a Pyramid Pyramid: polyhedron in whose base is a

polygon and the lateral faces are triangles with a common vertex.

The intersection of two lateral faces is a lateral edge.

The intersection of the base and lateralface is a base edge.

The altitude, or height, is the perpendicular distance between the base and vertex.

Regular pyramid – pyramid whose base is a regular polygon and the lateral faces are congruent isosceles triangles

Slant height (l) – length of the altitude of a lateral face of the pyramid

Lateral Area of a Pyramid Lateral Area (LA) – of a pyramid is

the sum of the areas of the congruent lateral (triangular) faces

The lateral area of a regular pyramid is half the product of the perimeter (p) of the base and the slant height (ℓ) of the pyramid.

PAL2

1..

Surface Area of a Pyramid

BPASorBALAS 2

1......

SA = area of base + 4(area of triangles)SA = B + 4( ½ bh)SA = B + ½ (4b) h → B + ½ Pℓ

Surface Area (SA) – of a pyramid is the sum of the lateral area and the area of the base

The surface area of a regular pyramid is the sum of the lateral area and the area of the base (B).

B

net

Finding the Surface Area of a Cone

A circular cone, or cone, has a circular base and a vertex that is not in the same plane as the base.

The altitude, or height, is the perpendicular distance between the vertex and the base.

In a right cone, the altitude is the perpendicular segment from the vertex to the center of the base

Lateral and Surface Areas of a Cone

Lateral Area: of a right cone is half the product of the circumference (C) of the base and the slant height (ℓ )of the cone.

LA = ½ C ℓ = πr ℓ

Surface Area: of a cone is the sum of the lateral area and the area of the base.

SA = LA + B or SA = ½ C ℓ + πr2 = πr ℓ + πr2

Volume of a Prism

Volume of a Cylinder

Composite Space Figures

A three-dimensional figure that is the combination of two or more simpler figures.

You can find the volume of a composite space figure by adding the volumes of the figures that are combined.

Volume of a Pyramid:

Volume of a Cone:

r

h

r

h 31

21

3Volume r h

Circle: set of all points in a plane that are a given

distance (radius) from a point (center).

Sphere: set of all points in space that are a given distance (radius) from a point (center).

Chord of a sphere: segment whose endpoints are on the sphere.

Diameter: chord that contains the center.

.diameter

chord

Surface Area of a Sphere

Volume of a Sphere

Similar Solids – two solids with equal ratios (scale factor) of corresponding linear measures.

Not similarSimilar

Similar

Theorem 11-12Areas and Volumes of Similar Solids

If the scale factor of two similar solids is a:b, then,

the ratio of their areas is a2 : b2,the ratio of their volumes is a3 : b3.

Circle Terminology A CHORD of a circle is a segment that has its

endpoints on the circle:  Chord CD, chord GH, and chord EF.

A SECANT is a line that intersects a circle in exactly two points.  Every secant forms a chord. 

A secant that goes through the center of the circle forms a diameter.

A line is TANGENT to a circle if it intersects the circle in EXACTLY ONE point.

This point is called the POINT OF TANGENCY.

interior

exterior

Theorem 12-1If a line is tangent to a circle, then

the line is perpendicular to the radius at the point of tangency.

Theorem 12-2If a line in the plane of a circle is

perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

Theorem 12-4 and its ConverseTheorem: Within a circle or in

congruent circles, congruent central angles have congruent arcs.

Converse: Within a circle or in congruent circles, congruent arcs have congruent central angles.

Theorem 12-5 and its Converse

Theorem: Within a circle or in congruent circles, congruent central angles have congruent chords.

Converse: Within a circle or in congruent circles, congruent chords have congruent central angles.

Theorem 12-6 and its Converse

Theorem: Within a circle or in congruent circles, congruent chords have congruent arcs.

Converse: Within a circle or in congruent circles, congruent arcs have congruent chords.

Theorem 12-7 and its Converse

Theorem: Within a circle or in congruent circles, chords equidistant from the center or centers are congruent.

Converse: Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers).

Theorem: In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

Theorem: In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord.

Theorem: In a circle, the perpendicular bisector of a chord contains the center of the circle.

Theorem 12-3If two tangent segments to a circle

share a common endpoint outside the circle, then the two segments are congruent

VocabularyInscribed Angle: an angle whose

vertex is on the circle and whose sides are chords of the circle

Intercepted Arc: an arc with endpoints on the sides of an inscribed angle

Inscribed Angle TheoremThe measure of an inscribed angle is

half the measure of its intercepted arc.

Corollaries to The Inscribed Angle Theorem

Corollaries to The Inscribed Angle Theorem

Theorem 12-12: The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Theorem 12-13

The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs.

Theorem 12-14The measure of an angle formed by two

lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.

Theorem 12-15

For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle.

Theorem 12-15: CASE I

The products of the chord segments are equal.

Theorem 12-15: CASE IIThe products of the secants and their

outer segments are equal.

Theorem 12-15: CASE III

The product of a secant and its outer segment equals the square of the tangent.

You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.

The radius of a circle is r and the center is (h, k).

Theorem 12-16: Equation of a Circle

Note: If the center is the origin, then the standard equation is x 2 + y

2 = r 2.

(x – h) 2 + (y – k)

2 = r 2

x

y

(h, k)

(x, y)

r

The standard equation of a circle with center (h, k) and radius r is