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
Thursday, November 30, 2006
Wednesday, November 29, 2006
notes 8-5
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 ~
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 ~
Tuesday, November 28, 2006
notes 8.4 and 8.3
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
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
notes 8-1 and 8-2
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
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
Tuesday, November 21, 2006
Monday, November 20, 2006
notes 6-5 (6-6 and 6-7 are on the poster sheet)
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
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
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
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
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
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
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
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
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
||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
Monday, November 13, 2006
Balancing Shapes Project!!!!!!!!!!
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!
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
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
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
Wednesday, November 08, 2006
hw: 11/8
In honor of my little brother's birthday NO HOMEWORK! Actually, no homework because you took a midterm today. Enjoy the night off!
Tuesday, November 07, 2006
Monday, November 06, 2006
Friday, November 03, 2006
Thursday, November 02, 2006
Wednesday, November 01, 2006
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