Geometry

We had a good time!!!!

2007-01-21T13:10:00.000-08:00

Here is that poem for anyone that wants it:

So many things have happened since September.
How many of them will you remember?
Will it ever come up somewhere in your mind
The stories I told of my sister’s Jeopardy time?
Or how about the con-men in the class?
They were trying to see if the answer was SAS.
If you ever feel like giving up and living underground,
At least listen for my metal detector sound.
For it would surely go off if it came near to you
Because you all have hearts of gold, it’s true!
Your life will be filled with many adventures.
Like this time when my aunt flushed away her new dentures!
You never know what is around any bend.
It could be a lawnmower in need of a mend!
If I didn’t send you off with some good advice I would be amiss
So if you will do anything for me it is this,
Do not ever limit yourself to what is easy
For knowing what you will miss will make you queasy
I do not mean to sound like a fortune cookie
Just please trust my experience, I’m no rookie!
So off you go and I hope you go glad
I know I will because this class wasn’t half-bad. ♥

hw: 1/11

2007-01-11T11:45:00.001-08:00

hw: finish ws on ch 12

2007-01-11T11:43:00.000-08:00

12.2 Surface Area of Prisms and Cylinders

Surface area of a right prism

Th. 12.2: S=2B+Ph S= Surface area of a right prism

B=Area of ONE of the bases

P= Perimeter of ONE base

H= height

Surface area of a right cylinder

Th. 12.3: S=2B + Ch S= Surface area of a right cylinder

B=Area of ONE of the bases

C= Circumference of ONE base

H= height

12.3 Surface Area of Pyramids and Cones

Surface area of a regular pyramid

Th. 12.4: S=B+ ½ Pl S= Surface area of a regular pyramid

B=Area of the base

P= Perimeter of the base

l= slant height

Surface area of a right cone

Th. 12.5: S= пr² + пrl S= Surface area of a right cone

r=radius of the base

l= slant height

12.4 Volume of Prisms and Cylinders

Volume of a cube

Postulate 27: V=s³ V= Volume of a cube

s= length of a side

Volume of a Prism

Th. 12.7: V=Bh V= Volume of a prism

B=Area of the base

h= height

Volume of a Cylinder

Th. 12.8: V=Bh V= Volume of a cylinder

B=Area of the base

h= height

12.5 Volume of Pyramids and Cones

Volume of a pyramid

Th. 12.9: V= 1/3 Bh V= Volume of a pyramid

B=Area of the base

h= height

Volume of a cone

Th. 12.10: V= 1/3 Bh V= Volume of a cone

B=Area of the base

h= height

12.6 Surface Area and Volume of Spheres

Surface Area of a sphere

Th. 12.11: S= 4пr² S= Surface area of a sphere

r=radius

Volume of a sphere

Th. 12.12: V= 4/3 пr³ S= Surface area of a sphere

r=radius

hw: 1/10

2007-01-10T12:11:00.001-08:00

hw: ws 12-1

notes: 12.1

2007-01-10T12:09:00.000-08:00

12.1: Exploring Solids
Polyhedron: solid that is bound by polygons, called faces (think of gold bars) plural: polyhedra or polyhedrons
An edge is where two polygons meet
Vertex- point where 3 or more edges meet
Ex: prism
Pyramid

Other solids: sphere, cone, cylinder

Regular polyhedra have all faces congruent, convex if any 2 points can be connected by a line within the polyhedron

Platonic solids: (named after Plato) (regular polyhedrons)
-tetrahefron (4 faces)
-cube (6 faces)
-octahedron (8)
-dodecahedron(12)
-icosahedron(20)

Euler’s Theorem: The # of faces F, vertices V, and edges E of a polyhegron are related by: F+V=E+2

Pg 723 10-52
Ws 12-1

hw: 1/8

2007-01-08T12:48:00.002-08:00

hw: study for QUIZ TOMORROW ON CH 11 50PTS!!!

notes 11-5

2007-01-08T12:48:00.001-08:00

11.5 Areas of Circles and Sectors
Th. 11.7: Area of a Circle= pi*r2
A sector of a circ is the region bound by 2 radii (a slice)

Th 11.8: Area of a sector: The ratio of the area A of a sector of a circ to the area of the circ is = to the ratio of the measure of the intercepted arc to 360.
(A / pi*r2 ) = ( m arc AB/ 360 )

16” pizza cut into 8 = slices, what is the area of 1 slice?

Square with sides of 10 inches with 4 circs inside, find the area of the shaded region

Inscribe a pentagon into a circ, find the m of the shaded region if the length of the radii is 8 ft

Prac ws 1-7

notes 11-4

2007-01-04T12:06:00.002-08:00

11.4 Circumference and Arc Length
Theorem 11.6: Circumference of a circle C=2*pi*r

Problem******** the C of a circle is 100 ft. what is the distance from pt a to b if arc ab= 40 degrees????? Solve with corollary

-Corollary: The ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360.
Arc length of AB = measure of arc AB
2*pi*r__________________ 360

ws 11-4

hw: 1/4

2007-01-04T12:03:00.000-08:00

hw: pg 686-688 15-41 odds #42,43

hw: 1/3

2007-01-03T14:28:00.000-08:00

hw: finish ws from class (11-3)
extra credit hw: pg 679 #7-21

notes 11-3

2007-01-03T14:26:00.000-08:00

11.3 Perimeters and Areas of Similar Figures:
Theorem 11.5: Areas of Similar Polygons:
If 2 polys are similar with the lengths of corresponding sides in the ratio a:b, then the ratio of their areas is a2:b2
Side length of Quad 1 = a Side length of Quad 2=b (Area of Quad 1 /Area of Quad 2)= a^2/b^2

notes 11.1-11.2

2007-01-02T14:16:00.002-08:00

Chapter 11- Area of Polygons and Circles:

11.1- Angle measures in Polygons
Theorem 11.1: The sum of the measures of the interior angles of a convex n-gon is (n-2) * 180. (this can also be found by dividing the polygon into triangles by connecting vertices and adding 180 for each triangle formed)
Theorem 11.2- Polygon Exterior Angles Theorem:
The sum of the exterior angles of a convex polygon, one angle at each vertex, is 360

Ws 11-1

11.2- Areas of Regular Polygons (*Remember regular means equal sides, equal angles)
Theorem 11.3- Area of an Equilateral Triangle: A= 1/4 3 s2

The apothem of a polygon is the distance from its center to any side
Apothem shown 

Theorem 11.4: Area of a Regular Polygon: A= Area, a= length of the apothem and P= Perimeter
A= 1/2 aP

Ws 11-2

hw: 1/2

2007-01-02T14:16:00.001-08:00

hw: ws 11-2

hw: 12/20

2006-12-20T11:37:00.000-08:00

hw: circle puzzles (bigger and biggest)

notice: test ch 10

2006-12-20T05:13:00.002-08:00

Wednesday and Thursday this week we will be reviewing for the upcoming test on Ch 10 this Friday 12/22

hw: 12/19

2006-12-20T05:13:00.001-08:00

hw: practice problems worksheet

notes 10.5 and 10.6

2006-12-20T05:12:00.000-08:00

10.5 Segment Lengths In Circles
Theorem 10.15: 2 chords intersect in a circle. The product of the segments of one chord are equal to the product of th segments of the other chord
Theorem 10.16: 2 secants with the same endpoint. The product of the external and internal segments of a secant is equal to the product of the other secants segments.
Theorem 10.17: If a secant and a tangent segment share a point outside the circle, the product of the segments of the secant is equal to the tangent segment squared.

Practice

10.6 Equations of Circles

r= radius (h,k) is the coordinate of the center (x,y) is any pt on the circ.

sq root((x-h)2 + (y-k)2) = r or in standard form (x-h)2 + (y-k)2 = r2
(all 2's should be exponents!!!!!!!!!)
If the center of the circle is the origin, the equation is x2 + y2 = r2

practisimo

hw: 12/18

2006-12-20T05:11:00.000-08:00

hw: Assignments found at the beginning of each section 10.1-10.4 on the handout

notes 10.1-10.4

2006-12-20T05:10:00.000-08:00

Unfortunately, when I post these notes, all of the pictures are destroyed. The text, however, is uncompromised so here is a link to the notes but please see me of refer to your book for picture clarification.
http://docs.google.com/View?docid=d7j72pd_24crn75k

hw: 12/15

2006-12-20T05:06:00.001-08:00

hw: None, test today on Ch 9

hw: 12/13

2006-12-20T05:05:00.001-08:00

hw: ch 9 review in book

notes 9-6

2006-12-20T05:04:00.001-08:00

9.6 solving right triangles
for an acute angle if sin A= x, then sin-1x= m angle A
activity pg 567

ex 1-3 pg 568-569
practice drawing triangles from book

ws 9-6

hw: 12/12

2006-12-12T13:25:00.001-08:00

hw: pg 562 # 10-39

hw: 12/11

2006-12-12T13:24:00.000-08:00

hw: finish challenge ws and quiz corrections

notes 9-5

2006-12-12T13:23:00.000-08:00

9.5 Trigonometric Ratios
Trig ratio is the ratio of 2 sides of a RIGHT triangle
Trigonometry means measurement of triangles (Tri= 3)
3 basic trigonometric ratios:
-Sine of angle A= sinA= opp/hyp

-Cosine of angle A= cosA= adj/hyp

-Tangent of angle A= tanA= opp/adj

SOHCAHTOA, SOHCAHTOA, SOHCAHTOA

Show how to find a missing side given a right triangle, one angle and one side

Find an angle with only the sides given: Check on calc’s mode should be set on DEGREE

Guided practice pg 562

Clinometer activity
-Regroup and discuss

pg 562 #10-39

notes 9-4

2006-12-07T12:27:00.001-08:00

9-4 Special Right Triangles
th.9-8: In a 45,45,90 tri, the hypot is √2 times as long as each leg
th. 9-9: in a 30-60-90 the shortest leg is x, the hypot is 2x and the longer leg is √3x

hw: 12/7

2006-12-07T12:26:00.000-08:00

hw: special rt triangle ws

notes 9-1, 9-2, and 9-3

2006-12-06T11:33:00.000-08:00

9-1 Similar right triangles
Theorem 9-1: If the altitude is drawn to the hypot of a RT tri, then the 2 tris formed are ~ to the orig tri, and to each other

Th. 9-2: If an alt is drawn to the htpo of a rt tri, the length of the altitude is the geometric mean of the lengths of the 2 segments now composing the hypot

Th. 9-3 If an alt is drawn to the hpot of a rt tri, the length of each leg of the rt tri is the geometric mean of the length of the hypot and the segment of the hypot that is adjacent to the leg
Draw diagram from pg 529

In other words:
In these 2 new tris formed, the altitude will the the longer leg of 1 and the shorter leg of the other. It becomes the geometric mean of the other 2 sides (see diagram pg 529)
Same goes for the other sides

Try pg 531 #19-29 odd

9-2 Pythagorean Theorem
RIGHT TRIANGLES ONLY ----C IS ALWAYS THE HYPOT
a2+ b2= c2
try 1000 examples

9-3 Converse of Pythagorean Theorem
If a2+ b2= c2 , then the tri is a rt tri
If a2+ b2>c2 , then the tri is acute
a2+ b2< c2, then the tri is obtuse

hw: 12/6

2006-12-06T11:32:00.001-08:00

hw: ws 9-1, 9-2, and 9-3 all ODDS plus #10 on ws 9-1

hw: 12/5 None

2006-12-05T12:34:00.000-08:00

Nice job on the test today, everyone passed!!!!!!!!!

hw: 12/4

2006-12-04T11:16:00.001-08:00

Study for Test tomorrow on ch 8!!!!!!!!!

notes 8-7

2006-12-01T11:36:00.000-08:00

8.7 Dilations
Dilation: when the image and preimage of a figure are proportional

A dilation with center C and scale factor, k, is a transformation that maps every point P in the plane to a point P′ and has the following properties:
1) If P is not the center point C, then the inage point lies on ray CP. The scale factor k is a positive number such that k = (CP′/CP), and k is not equal to 1
2) If P is the center point C, then P = P′

A dilation is a reduction if 01

Draw diagram pg 506

Use overhead to show dilation in action

Ex 1 pg 506

Ex 2 pg 508

Construction: drawing a dilation pg 507

Real life app 8-7

Ws 8-7

notes 8-6

2006-11-30T11:42:00.000-08:00

8-6 Proportions and Similar Triangles
Theorem 8-4: Triangle Proportionality Theorem: If a line || to one side of a tri intersects the other 2 sides, then it divides those sides proportionally.

8.5 Converse: If a lines divides 2 sides of a tri proportionally then it is || to the third side

8.6: If 3 || lines intersect 2 transversals, then they divide the transversals proportionally

8.7: If a ray bisects an < of a tri, then it divides the opp. Side into segments whose lengths are proportional to the other 2 sides

hw: 11/30

2006-11-30T11:40:00.000-08:00

hw: ws 8-6

notes 8-5

2006-11-29T11:41:00.001-08:00

8.5 Proving Triangles are Similar
Theorem 8-2 : SSS Similarity: If the corr. Sides of 2 tris are proportional, then the tris are ~

Theorem 8-3: SAS Similarity: If 1 angle on 1 tri is cong to an angle of a 2nd tri and the lengths of the sides including these angles are proportional, then the tris are ~

hw: 11/29

2006-11-29T11:37:00.000-08:00

hw: ws 8-5

hw: 11/28

2006-11-28T11:45:00.000-08:00

hw: finish pg 484 24-46evens and pg 476 11-41 odds

notes 8.4 and 8.3

2006-11-28T11:36:00.002-08:00

8.4 Similar Traingles
SIMILAR: same shape, different size
AASimilarity Postulate: If 2 angles of 1 tri are cong to 2 angles of another tri, then the 2 tris are similar
-Activity pg 480 need protractors
Why do lines only have 1 slope? Why can you pick ANY 2 pts on a line and still get the same slope?
Scale factor: the ratio of the length of 2 corresponding sides
Ex 5 pg 482 (only ½ the tri’s)

8.3 Similar Polygons
- If corresponding angles of 2 polys are cong, and the lengths of the corresponding sides are proportional
“~” means similar
- Must name ~ polys correspondingly

Ex 1-3 pg 473

Theorem 8.1: If 2 polys are ~, then the ratio of their perimeters is = to the ratios of the corresponding side lengths

Ex 4 pg 474

Pg 484 24-46 evens, pg 476 11-41 odd

hw: 11/27

2006-11-28T11:36:00.001-08:00

hw: ws 8.1 and 8.2

notes 8-1 and 8-2

2006-11-28T11:35:00.000-08:00

Mobiles Due in 1 week!

8.1 Ratios and Proportions and 8.2 Prob Solving in Geo with Proportions:
Ratio: 2 quantities in the same units being compared. The ratio of a to b is a/b or a:b. *b cannot be zero *ratios usually in simplest form

Proportions: 2 ratios that are equal to each other.
Like 50/100=1/2
1. Cross Product Prop
If a/b is = to c/d then you can write a/b=c/d. a and d are known as the extremes and b and c are the means. If you multiply the 2 extremes together and the 2 means together, those products are called the cross products which are equal to each other so ad=bc

2. Reciprocal Prop
The reciprocals of both ratios will also be equal.
Practice solving proportions

------8.2

3. if a/b= c/d then a/c= b/d

4. if a/b= c/d then (a+b)/b = (c+d)/d

ex 3-4 pg 458, 5 pg 459, ex 2-3 pg 466

ws from comp (mc too)

ws 8-1 ws 8-2

2006-11-21T12:26:00.000-08:00

11/22 HAPPY THANKSGIVING!!

2006-11-21T12:10:00.000-08:00

hw: 11/21 None- Test today

2006-11-21T11:43:00.000-08:00

hw: None- Test today

notes 6-5 (6-6 and 6-7 are on the poster sheet)

2006-11-20T11:44:00.001-08:00

6.5: Trapeziods and Kites
Trapeziod: Quad with exactly one pair of || sides, called the bases. This creates 2 pairs of base angles. Non-|| sides are called legs. If those legs are cong, it is an isosceles trapezoid.
Midsegment: connects the midpoints of the legs

Theorems:
6.14: If a trap is isosceles, then each pair of base angles is cong
6.15: If a trap has a pair of cong base angles, then it is an isos trap
6.16: A trap is isos. If and only if its diagonals are cong.
6.17: Midsegment Theorem for Trapezoids: The midsegment of a trapezoid is || to each base and its length is one half the sum of the lengths of the bases.

Kites: quad with 2 pairs of consecustive cong sides, but opp sides are NOT cong

Theorems for Kites:
6.18: If a quad if a kite, then its diagonals are perpendicular
6.19: If a quad if a kite, then exactly on pair of opposite angles are cong.

Guided prac pg 359

notes 6-3 and 6-4

2006-11-20T11:43:00.001-08:00

6.3 Proving Quads are ||ograms
Theorems:
6.6: If both pairs of opposite sides of a quad are cong, then the quad is a ||ogram
6.7: If both pairs of opp angles of a quad are cong, then the quad is a ||ogram
6.8: if an angle of a quad is supplementary to both of its consecutive angles then the quad is a ||ogram
6.9: If the diagonals of a quad bisect each other, then the quad is a ||ogram
6.10: if one pair of opposite sides of a quad are cong and || then the quad is a ||ogram

To prove quads are ||ograms, show that:
• both pairs of opp sides are ||
• both pairs of opp sides are cong
• both pairs of opp angles are cong
• one angle is supplementary to both its neighbors (consec angles)
• the diagonals bisect each other
• one pair of opposite sides are congruent and ||

*In coordinate geometry you can use the slopes and side lengths to also prove figures to be ||ograms*
guided prac pg 342

6.4 Rhombuses, Rectangles, and Squares

venn diagram sheet

Rhombus: ||ogram with 4 cong sides
Rectangle: ||ogram with 4 right Square: ||ogram with 4 cong sides and 4 right angles

Corollaries:
-Rhombus: A quad is a rhombus if and only if it has 4 cong sides
-Rectangle: A quad is a rectangle if and only if it has 4 right -Square: A quad is a square if and only if it is a rhombus and a rectangle
Theorems:
6.11: A ||ogram is a rhombus if and only if its diagonals are perpendicular
6.12: A ||ogram is a rhombus if and only if each diagonal bisects a pair of opposite rays
6.13: A ||ogram is a rectangle if and only if its diagonals are congruent.

Guided prac pg 351

notes 6-2

2006-11-20T11:42:00.000-08:00

6.2 Properties of Parallelograms
||ograms are quad with both pairs of opposite sides ||

Theorem 6.2: If a quad is a ||ogram, then its opposite sides are congruent
6.3: If a quad is a ||ogram then its opposite angles are cong
6.4: if a quad is a ||ogram, then its consecutive angles are supplementary
6.5: if a quad is a ||ogram, then its diagonals bisect each other

Balancing Shapes Project!!!!!!!!!!

2006-11-13T12:47:00.000-08:00

Balancing Shapes Project!!!!!!!!!!

Due Date: 12/4

Objective: Explore the balancing points of triangles and other shapes

Directions: Using your knowledge of geometry so far, you can use certain line properties to find where specific shapes will balance.
Use pages 316-317 in your book as a guide. You will be creating a hanging mobile using geometric shapes and their balancing points.

Guidelines:
1) Your mobile must be able to hang freely
2) You must use at LEAST one square, rectangle, parallelogram, rhombus, and circle that are hung from their balancing points (see diagram pg 317)
3) You may use any additional shapes
4) You may use drinking straws, thread, sticks, dowels, cardboard, twine, index cards, or any other type of materials you choose but remember it must be light enough to hang from the ceiling (no bricks please!)
5) As we are entering a festive season, feel free to add a seasonal touch to your mobile, though this is not required.
6) Creativity is encouraged!

notes 6-1

2006-11-13T12:46:00.000-08:00

6-1 Polygons
1) Formed by 3 or more sides where the common endpoints are not collinear
2) each side intersects exactly 2 other sides, one at each endpoint (vertex)
Naming: Use the vertices in consecutive order

Draw shapes are they polygons?

# of sides Name of polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n-gon

Convex polygon: no line that contains a side has points that lie inside the polygon
Concave: line containing a side DOES contain points on the interior

Draw shapes concave? Convex?

Equilateral: all sides cong
Equiangular: all angles cong
Regular: when the poly is equiangular and equilateral
Diagonal: Segment that joins two nonconsecutive vertices

Theorem 6.1: Interior angles of a Quadrilateral:
The sum of the measure of the interior angles of a quad is 360

Draw poly: angles 80, 70, 2x, x- find angles

Ws 6-1

hw: 11/8

2006-11-08T13:36:00.000-08:00

In honor of my little brother's birthday NO HOMEWORK! Actually, no homework because you took a midterm today. Enjoy the night off!

hw: 11-7

2006-11-07T13:16:00.001-08:00

hw: studystudystudy midterm tomorrow!!!!!!!

hw: 11-6

2006-11-06T10:48:00.000-08:00

hw: review packet and STUDY FOR MIDTERM!!! (it's on Wedsnesday!)

hw: 11/3

2006-11-03T13:37:00.001-08:00

hw: study for midterm WEDNESDAY 11/8

hw: 11-2

2006-11-02T13:42:00.001-08:00

hw: study for test tomorrow on Ch 5 REMEMBER TO PAY SPECIAL ATTENTION TO YOUR STUDY SHEET

hw: 11/1

2006-11-01T12:40:00.001-08:00

hw: review packet- TEST MOVED TO FRIDAY

notes: 5-6

2006-10-31T11:16:00.000-08:00

5-6 Indirect Proof and Inequalities in 2 Triangles
Indirect Proof: assume the opposite is true. When you are unable to prove the opposite, you know the original statement is true.
Do ex 1 pg 302

Hinge Theorem: If 2 sides of 1 tri and cong to 2 sides of another tri, and the included angle of the 1st is larger than the included angle of the 2nd, then the third side of the first is longer than the third side of the 2nd
Converse: If 2 sides of 1 tri are cong to 2 sides of another tri, and the third side of the 1st is longer than the 3rd side of the second, then the included ang of the first is larger than the included angle of the second.

hw: 10/31 HAPPY HALLOWEEN!

2006-10-31T11:15:00.000-08:00

HW: none, extra credit picture if you like

Weeks Schedule 10/30 - 11/7

2006-10-30T11:55:00.000-08:00

10/30- section 5-5
10/31- 5-6
11/1- review ch 5
11/2- test ch 5
11/3- begin review for midterm *grades close*

11/6- review for midterm
11/7- MIDTERM

notes 5-5

2006-10-30T11:54:00.000-08:00

5-5 Inequalities in one Triangle
The largest angle in a triangle will be opposite the longest side
The smallest angle will be opposite the smallest side

Theorem 5.10: If one side of a tri is longer than another side, then the angle opposite the longer side is larger that the angle opposite the shorter side

5-11: If one angle of a tri is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle

5-12 : Exterior Angle Inequality: The measure if an exterior angle of a tri is greater than the measure of either of the 2 nonadjacent interior angles.

5-13: Triangle Inequality: The sum of the lengths of any 2 sides of a triangle is greater then the length of the third side

ws one tri ineq
flip 5-5 ws hw

hw: 10-30

2006-10-30T11:53:00.000-08:00

hw: finish worksheet 5-5 and do corrections from quiz on a seperate sheet

Notes 5-4

2006-10-26T11:28:00.001-07:00

5-4 Midsegments
A midsegment connects midpoints of a triangle.

Midsegment Theorem: The segment connecting the midpts of 2 sides of a triangle is parallel to the third side and is ½ as long.

hw: 10/26

2006-10-26T11:19:00.001-07:00

hw: ws 5-4

hw: 10/25

2006-10-26T11:18:00.000-07:00

hw: ws 5-3

Notes 5-3

2006-10-26T11:17:00.000-07:00

5-3 Medians and Altitudes
Median: from the vertex to the midpt of the opposite side
• 3 medians meet at the centroid of the triangle.

Concurrency of Medians of a triangle Theorem: The medians of a triangle intersect at a pt that is 2/3 the distance from each vertex to the midpt of the opposite side.

*Centroid can be used as a balancing pt

do ex’s from tracys sheet

Altitude: seg perpendicular to base from a vertex
• Altitudes meet at the orthocenter (different from acute, rt, and obtuse triangles)

Concurrency of Altitudes of a triangle Theorem: The lines containing the altitudes of a triangle are concurrent

Circumcenter: pt of concurrency of perp bis of a triangle
*distance from vertices are cong.
Incenter: pt of concurrency of < bisectors of a triangle
• pt is equidistant from the sides of the triangle
Centroid: pt of concurrency of medians of a triangle
• pt is 2/3 distance from vertex to side
Orthocenter: ot of concurrency of altitudes of a triangle
*nothing special

Notes 5-2

2006-10-24T12:29:00.002-07:00

5-2 Bisectors of a Triangle
Perp Bis of a triangle: a line or ray or segment that is perp to a side at the midpt

-Triangle cutout activity: fold to make perp bis of each side: do for each side, notice anything?

Concurrent lines: when 3 or more lines intersect at 1 pt, called the pt of concurrency

• The pt of concurrency of the perp bis of a triangle is called the circumcenter of the triangle

Concurrency of perp bis of a triangle Theorem: The perp bis of a triangle intersect at a pt equidistanct from the vertices of a triangle

< bisector of a triangle – bisects and <

Concurrency of < bis of a triangle Theorem: The < bis of a triangle intersect at a pt that is equidistant fromt eh sides of the triangle

hw: 10/24

2006-10-24T12:29:00.001-07:00

hw: ws 5-2

HW: 10/23

2006-10-23T11:17:00.000-07:00

hw: ws 5-1

Notes 4-7

2006-10-16T12:08:00.003-07:00

4.7 Triangles and Coordinate Proof
CP involves places geometric figures in a coordinate plane and using distance formulas, midpoints, etc to prove statements

Tri with vertices at (0,0) (12.0) and (12, 5) we can find the distances quite easily
D=
Find the midpoint of a line:
((x1+x2)/2 , (y1+y2)/2)

do ex 4 on pg 245

hw: 10/16

2006-10-16T12:08:00.001-07:00

hw: ws on triangles

Notes 4.6

2006-10-12T11:12:00.002-07:00

4.6 Isosceles, Equilateral, and Right Triangles
draw base angles and vertex angle legs and base on isosceles tri

Th. 4.6: Base < s Theorem: If 2 sides of a tri are cong., then the angles opposite them are cong.
Corollary: If a tri is equilateral, then it is equiangular
Th 4.7: Converse of the Base < theorem: If 2 < s of a tri are cong, then the sides opposite them are cong
Corollary: If a tri is equiangular, then it is equilateral
The. 4-8 Already have, HL

Pg 241 33 and 34

Ws 4-6

hw: 10-12

2006-10-12T11:12:00.001-07:00

hw: ws 4-6

quiz tomorrow first part of ch 4
test on ch 4 probably tues or weds next week!

Notes 4-5

2006-10-11T12:00:00.002-07:00

No real "notes" on this seciton, follow along the following proofs:

Ex. 2 on pg 213- SAS
Ex. 2 pg 221- ASA
#19 pg 224- AAS
ex 1 pg 212- SSS

hw: 10-11

2006-10-11T12:00:00.001-07:00

hw: ws 4-5

Notes 4-1 and 4-2

2006-10-06T11:55:00.000-07:00

4-1 Triangle Congruence
Classification by sides
Equilateral Triangle
An equilateral triangle has got 3 sides of equal length and 3 angles that are equal.
Since ALL the angles in a triangle add up to 180º then 180 divided by 3 must be 60º.
Isosceles TriangleAn Isosceles triangle has got two sides of equal length and 2 angles equal.
Scalene TriangleA scalene triangle is the easiest of them all. The scalene triangle has got NO sides of equal length and NO angles the same.

Classification by angles

Acute Triangle
All angles are less than 90°

Right Triangle
Has a right angle (90°)

Obtuse Triangle
Has an angle more than 90°
Equiangular: triangle with 3 congruent angles

Triangles can fit into more than one category:
Be as specific as you can: angles of 58, 65, and 57 make an acute scalene triangle
< 130 and 2 congruents sides make and obtuse isolsceles
Each “corner” of the triangle is called a vertex (pl. vertices)
Two sides sharing a common vertex are called adjacent sides

Right : Sides that form the rt angle are legs, the opposite side, or the longest side of the triangle is the hypotenuse
and Isosceles: if only 2 sides are congruent, they are called legs, the other side is the base

Corollary- statement easily proved by a theorem

Theorem 4.1 Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle= 180
Corollary: The acute angles of a right triangle are complementary
Theorem 4.2: Exterior angle Theorem: The m of an ext. angle of a triangle is = to the sum of the m of the 2 nonadjacent interior angles
Draw pic

4.2 Congruence and Triangles
Congruent: Same size and shape
Corresponding angles and sides are cong.

Theorem 4.3: Thrid angles Theorem: If 2 angles of a triangle are cong. To 2 angles of another triangle, the third angle is also cong.
T 4.4 Properties of Cong triangles
Reflexive: every tri is cong to itself
Symmetric: if ∆ABC is cong to ∆DEF, then ∆DEF is cong to ∆ABC
Transitive: if ∆ABC is cong to ∆DEF, and ∆DEF is cong to ∆JKL then ∆ABC is cong to ∆JKL

Ws angles and triangles
Ws 4-1 and 4-2

Notes 4-3 and 4-4

2006-10-06T11:53:00.000-07:00

4.3 and 4-4 Proving Triangles are Congruent: SSS, SAS, ASA, and AAS

Postulate 19: Side Side Side SSS Congruence Post: If 3 sides of one tri and cong to 3 sides of another tri, the tris are cong
Postulate 20: Side Angle Side SAS Congruence Post: If 2 sides and the included angle of one tri are cong to 2 sides and the included angle of another tri, the tris are cong
Postulate 21: Angle Side Angle ASA Congruence Post: If 2 angles and the included side of one tri are cong to 2 angles and the included side of another tri, the tris are cong
Postulate 22: Angle Angle Side AAS Congruence Post: If 2 angles and a nonincluded side of one tri are cong to 2 angles and the corresponding nonincluded side of another tri, the tris are cong

Ws’s

HW:10/3

2006-10-03T11:13:00.001-07:00

hw: I forgot to give it to you so enjoy the night off!

HW: 10/2

2006-10-02T11:37:00.001-07:00

hw: section 3-6 bookwork from sheet

Notes: 3-7

2006-10-02T11:36:00.000-07:00

3-7 Perpendicular lines in the Coordinate Plane
Postulate 18, Slopes of Perpendicular Lines: 2 nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vert and horiz lines are perpendicular
Do ex 1 -4 pg 172
practice ws

HW: 9/29

2006-09-29T11:55:00.000-07:00

HW: ws 3-6

Notes 3-6

2006-09-29T11:54:00.000-07:00

Practice with angles

3.6 Parallel Lines in a Coordinate Plane
Equation of a line (slope- intercept): y=mx+b
M= slope rise/run or m= (y2-y1)/(x2-x1)
B= y-intercept
(point-slope) y-y1= m(x-x1)

Find slope between (2,5) and (19, 8)
(-10, -4) and (12, -9)

What is the slope of a train that goes 4 ft higher for every 10ft it goes forward?

What is the slope of the following line: y= -1/2 x – 5

Postulate 17 Slopes of || lines: In a coordinate plane, 2 non vertical lines are || if and only if they have the same slope. Any 2 vertical lines are ||

Would lines through the following points be ||?
(0, 6) and (2,0); (-2, 6) and (0, 1); and (-6, 5) and (-4, 0)

write an equation through the point (2,3) that has a slope of 5

line 1 has the equation y= -1/3 x -1
write the equation of the line || to line 1 that passes through the point (3,2)

HW: 9-28

2006-09-28T15:00:00.000-07:00

None, as we retook the quiz we did not have enough time to get through the next section!

Notes 3-5

2006-09-27T14:46:00.000-07:00

3.5 Using Properties of Parallel Lines
Theorem 3.11: If 2 lines are || to the same line, then they are || to each other
Theorem 3.12: In a plane, if 2 lines are perpendicular to the same line, then they are || to each other
#Constructions: pg 159; copying an angle, || lines

construction practice ws

HW: 9-27

2006-09-27T14:45:00.000-07:00

HW: section 3.5 on assignment sheet

Notes 3-4

2006-09-26T13:39:00.000-07:00

Quiz
Transversal and corres. angles ws

3.4 Proving lines are Parallel
How can you prove 2 lines are ||?????????? All the theorems we learned yesterday are useless if you cannot prove that the lines are || to begin with. Here’s how you do it:
If a transversal cuts 2 lines and the ____________________ . then the lines are ||
Postulate 16 Corresponding angles Converse- corresponding angles are congruent
Theorem 3.8 Alternate Interior (AI) Converse- AI angles are congruent
Theorem 3.9 Consecutive Interior (CI) Converse- CI angles are supplementary
Theorem 3.10 Alternate Exterior (AE) Converse- AE angles are congruent

Ex 1-3. proving about theorems
Try pg 155 # 30 then practice 153 1-9

Ws 3-4

HW: 9-26

2006-09-26T13:38:00.000-07:00

HW: section 3.4 from the assignment sheet

Reminder- Quiz Tues 9/26 on 3-1 through 3-3

2006-09-25T14:10:00.001-07:00

Reminder- Quiz Tues 9/26 on 3-1 through 3-3

Notes 3-2 and 3-3

2006-09-25T14:09:00.001-07:00

3.2 Proof and Perpendicular Lines
proof- proof written as a paragraph
flow proof uses arrows instead of formal 2 columns ( ex 1 on pg 136)
Theorem 3.1: If 2 lines intersect to form a linear pair of s, then the lines are
Theorem 3.2: If 2 sides of 2 adjacent acute s are , then the s are complementary
Theorem 3.3: If 2 lines are , then they intersect to form 4 right angles.

Practice ws 3-2- they already have this on the back of 3-1

3.3 Parallel lines and Transversals
Postulate 15- corresponding Angles Post.: If 2 || lines are cut by a transversal, the pairs of corresponding angles are congruent
Theorem 3.4 Alternate Interior s: If 2 || lines are cut by a transversal, then the pairs of alternate interior s are congruent
Theorem 3.5 Consecutive Interior s: If 2 || lines are cut by a transversal, then the pairs of consecutive interior s are supplementary
Theorem 3.6 Alternate Exterior s: If 2 || lines are cut by a transversal , then the pairs of alternate exterior s are congruent
Theorem 3.7 Perpendicular Transversal: If a transversal is perpendicular to one of 2 || lines, then it is perpendicular to the other

Examples 1-5 pg 144-145 (5 is particularly interesting)

|| and perpendicular lines
transversals and corresponding angles

HW: 9-25

2006-09-25T14:08:00.000-07:00

HW: pg 139-140 11-15; pg 146-148 18-28

Notes: 3.1

2006-09-25T05:10:00.000-07:00

3.1 lines and Angles
parallel lines are coplanar and do not intersect
skew lines do not intersect and are not coplanar
parallel planes do not intersect
Postulate 13- Parallel Post. : If there is a line and a point not on the line, then there is exactly one line through the point that is parallel to the given line
Postulate 14- Perpendicular Postulate: If there is a line and a point not on the line, then there is exactly one line through the point that is perpendicular to the given line

*construction*- making a perpendicular to a line (pg 130)
-have them do the other constructions we know, bisecting a line/making a perpendicular, copying a line, bisecting an angle

transversal- line that intersects 2 or more coplanar lines
The angles a transversal creates have special names:
Corresponding angles: occupy corresponding positions
Alternate Exterior: lie outside the 2 lines on opposite sides
Alternate Interior: ;lie between the 2 lines on opposite sides
Consecutive Interior: between the 2 lines and on the same side (aka same side interior)

Dashed angle organizer
Ws: practice 3-1

HW: 9-22

2006-09-25T05:09:00.000-07:00

HW: pg 132-133; 10-26 AKA the section #1 assignment on the sheet

HW: 9-21

2006-09-21T12:05:00.001-07:00

HW: section 3-1 on sheet provided- HOWEVER this is not due until Monday

HW: 9-20 NONE!

2006-09-20T12:19:00.000-07:00

HW: 9-19

2006-09-19T11:03:00.000-07:00

Hw: pg 121-123 chapter test and standardized test

NOTICE: CH 2 TEST WEDNESDAY

2006-09-18T11:07:00.001-07:00

NOTICE: CH 2 TEST WEDNESDAY

Notes 2-6

2006-09-18T10:59:00.000-07:00

Intro to Proofs

2.6 Proving Statements about Angles
Theorem 2.2- Prop of Angle Congruence-
< Congruence is Reflexive (
Theorem 2.3- Right angle Congruence Theorem- All rt angles are congruent

Theorem 2.4- Congruent Supplements Theorem- If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.

Theorem 2.5- Congruent Complements Theorem- If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.

Postulate 12- Linear Pair Postulate- If two angles form a linear pair, then they are supplementary.

Theorem 2.6- Vertical Angle Theorem- Vertical Angles are Congruent

Pg 114 #23 and #24 together

Practice 2-6

Quiz: pg 116; 1-9

HW: 9-18

2006-09-18T10:57:00.000-07:00

HW: from assignment sheet: all problems listed for page 112, all problems EXCEPT 23 and 24 for page 114

HW:9-15

2006-09-15T13:58:00.000-07:00

HW: I did stray form what is on your sheet just a little:
For monday:
pg 93 45-48 ONLY
pg 99 all on the sheet EXCEPT 19 AND 23
pg 105 all listed on the sheet

SORRY!

2006-09-15T13:56:00.000-07:00

SORRY THAT I WAS LATE POSTING TODAY! HAD A FEW ERRANDS TO RUN RIGHT AFTER SCHOOL! SEE YOU AT THE FOOTBALL GAME!

Notes: 2-3 and 2-4 *Not all symbols appear correctly in notes

2006-09-14T12:43:00.000-07:00

2-3 Deductive Reasoning
CS can be written using notations
- “p” represents the hypothesis
- “q” if the conclusion
- “  “ is read as “implies”

CS: If the sun is out, then the weather is good. p q
p q
Converse: If the weather is good, then the sun is out q p
q p
BS: If p, then q, and if q then p OR p q OR p if and only if q
Try: p= the value of x is -5 q= the absolute value of x is 5
a. write p q in words
b. write q p in words
c. is pq true?
To write the inverse and contrapositive, you need to be able to negate the the statement symbolically. We use the (~) symbol to do so.
Statement: ∟3 measures 90 ∟3 is not acute
Symbol: p q
Negation: ∟3 does not measure 90 ∟3 is acute
Symbol: ~p ~q
The inverse of p q : ~p ~q If ∟3 does not measure 90, then ∟3 is acute
The contrapos of p q: ~q~p If ∟3 is acute, then ∟3 does not measure 90
For me: ex. 2 pg 88

Deductive Reasoning- uses facts, def., and properties in logical order to write and argument
Laws of Deductive Reasoning:
Law of Detachment: If pq is a true conditional statement and p is true, then q is true
(ex 4 on pg 89)
Law of Syllogism: if pq and qr are true conditional statements, then p r is true
Together ex 5 and ex 6,
prac 1-7 pg 91

2-4 Reasoning with Properties from Algebra
Addition Property: If a=b, then a+c= b+c
Subtraction: If a=b, then a-c=b-c
Multiplication: If a=b, thenac=bc
Division: If a=b and c not equal to 0, then a/c=b/c
Reflexive: For any real number a, a=a
-Geometric: segment AB= segment AB;;;;; m< A= m Symmetric: If a=b, then b=a
-Geometric: If AB=CD, then CD=AB;;;;; If m Transitive: If a=b, and b=c, then a=c
-Geo: If AB=CD and CD=ED, then AB=EF;;;; If m then m Substitution: If a=b, then a can be substituted for b in any equation or expression
Example 1 and 2 pg 96; 4 and 5 pg 98
Guided practice 1-9 pg 99

HW: 9-14

2006-09-14T12:42:00.000-07:00

HW: day 2 on assignment sheet

Notes end of 2-1, also 2-2

2006-09-13T11:22:00.000-07:00

Equivalent Statements: When 2 statements are both true or both false
Conditional statement (orig) and contrapositive
Inverse and Converse

Write the inverse, converse, and contrapositive:
If there is snow on the ground, then flowers are not in bloom.
If Micheal wins Project Runway, then I will be happy.

Point, line, and Plane Postulates:
Postulate 5: Through any two points there exists exactly one line
6: A Line contains at least 2 points
7: If two lines intersect, then their intersection is exactly one point
8: Through any 3 noncollinear points, there is exactly one plane
9: A plane contains at least 3 noncollinear points
10: If 2 points lie in a plane, then the line containing them lies in the plane
11: If two planes intersect, then their intersection is a line.

2-2 Definitions and Biconditional Statements:
Perpendicular- intersects at a 90 degree angle
Lines and planes can also intersect perpendicularly ( )
Biconditional statement- statement containing the words “if and only if” (essentially a conditional statement and its converse)
Ex. I will buy a mansion if and only if I win the lottery
CS: I will buy a mansion if I win the lottery
Converse: If I win the lottery I will buy a mansion.

Ex. b= 4 if and only if b2=16
CS: If b=4, then b2=16.
Converse: If b2=16, then b=4
The CS is true, but the converse is not. This makes the Biconditional Statement FALSE

When you can make a true CS and converse, you can make a true biconditional (true forwards and backwards)

Practice 2-2c