## Tuesday, July 24, 2007

### Geometry 9.4 Special Right Triangles

Geometry 9.4 Special Right Triangles:

I) 45°: 45°:90° Triangle Theorem: in a 45°, 45°, 90° triangle,
the hypotenuse is √2 times as long as each leg.
1x: 1x: x√2
Example: if the leg of a 45°: 45°: 90° triangle is 5, what is the other leg length and the hypotenuse length?
Since the legs are equal in length, the other leg is 5.
the hypotenuse would be 5√2

II) 30 :60 : 90 triangle theorem: in a 30º , 60º , 90º triangle,
the hypotenuse is 2 times as long as the shorter leg, and the longer leg is
√3 times as long as the shorter leg.
1x : x√3 : 2x
Example: if the shorter leg of a 30º , 60º , 90º triangle is 7, what is the length of the longer leg and the hypotenuse?
the longer leg is 7√3 and the hypotenuse is (2)(7) = 14

### Geometry 9.1 - 9.3 Similar Right Triangles, Pythagorean Theorem

Geometry 9.1 Similar Right Triangles:

I) Theorems:
A) if the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
B) In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.
The length of the altitude is the geometric mean of the lengths of the two segments.
C) In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Geometry 9.2 The Pythagorean Theorem:
I) Theorems:
A) Pythagorean Theorem: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
a2 + b2 = c2
Example: Given the two legs of a right triangle are 7 and 9, what is the hypotenuse?
72 + 92 = c2
49 + 81 = c2
130 = c2
c = √130 ≈ 11.40175425
B) Pythagorean Triple: is a set of three positive intergers, a, b, and c, that satisfy the equation
a2 + b2 = c2
Examples:
3, 4, 5
5, 12, 13
8, 15, 17
are a few

Geometry 9.3 The Converse of the Pythagorean Theorem:
I) Theorems:
A) Converse of the Pythagorean Theorem: if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
If c2 = a2 + b2, then Δ ABC is a right triangle.
B) If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.
if c2 is less than a2 + b2, then ΔABC is acute.
C) If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.
if c2 is greater than a2 + b2, then ΔABC is obtuse.
( you can think of this in a doorway, the length of the door does not change nor does the doorway. As you open the door, the angle formed between the door and the doorway gets larger, therefore the distance from the doorway to the end of the door not on the hinge (in the room) is getting longer. So if the angle formed from the intersection of the doorway and door makes a right angle, the triangle is right. If the angle formed is smaller than the right angle, the triangle is acute. If the angle formed is larger than the right angle, the triangle is obtuse.)
Examples:
1) Given a triangle has lengths of 2, 10, and 11 is it acute, right or obtuse?
22 + 102 = 4 + 100 = 104
112 = 121
therefore the c2 is longer so the triangle is obtuse.
2) Given a triangle has lengths of 6, 5 and 7, is it acute, right or obtuse?
52 + 62 = 25 + 36 = 61
72 = 49
therefore the c2 is smaller so the triangle is acute.
3) Given a triangle has lengths of 6, 8 and 10, is it acute, right or obtuse?
62 + 82 = 36 + 64 = 100
102 = 100
therefore the c2 is equal so the triangle is right.
4) Given a triangle has lenghts of 5, 6, and 11, is it acute, right or obtuse?
Recall that a triangle has to have the sum of two sides greater than the third side and since 5 + 6 = 11, these side lenghts cannot form a triangle so it is none of them.

### Geometry Chapter 6.2 - 6.6 Properties of Quadrilaterals

I) 6.2 Properties of Parallelograms
A) Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
B) theorems for Parallelograms:
If a quadrilateral is a parallelogram, then
1. both pairs of opposite sides are congruent.
2. both pairs of opposite angles are congruent.
3. both pairs of consecutive angles are supplementary.
4. the diagonals bisect each other.
5. a diagonal bisects the parallelogram into 2 congruent triangles
6. if one pair of opposite sides are parallel and congruent.

II) 6.3 Proving Quadrilaterals are Parallelograms:
1. Show that both pairs of opposite sides are parallel.
2. Show that both pairs of opposite sides are congruent.
3. Show that both pairs of opposite angles are congruent.
4. Show that one angle is supplementary to both consecutive angles.
5. Show that the diagonals bisect each other.
6. Show that one pair of opposite sides are congruent and parallel.

III) 6.4 Rhombuses, Rectangle, and Squares
A) Rhombus: A parallelogram with 4 congruent sides
1) the diagonals are perpendicular
2) the diagonals bisect opposite angles
B) Rectangle: A parallelogram with 4 congruent right angles.
1) the diagonals are congruent
C) Square: a parallelogram with 4 congruent sides and 4 congruent right angles.

IV) 6.5 Trapezoids and Kites
A) Trapezoid: A quadrilateral with exactly one pair of parallel sides called the bases. It has 2 pairs of base angles. The nonparallel sides are called the legs.
B) Isosceles Trapezoid: A trapezoid that has:
1) nonparalel sides are congruent
2) base angles are congruent
3) diagonals are congruent.
C) Kite: a quadrilateral with 2 pairs of consecutive congruent sides but the opposite sides are not congruent.
1) the diagonals are perpendicular
2) it has exactly one pair of opposite angles congruent. (the pair of angles that is between the noncongruent consecutive sides)
D) Midsegment of a trapezoid: the midsegment of a trapezoid connects the midpoints of the legs and is parallel to both bases. Its length is the average of the 2 bases.
Example: given trapezoid ABCD with BC ll AD and midsegment MN where M is the midpoint of AB and N is the midpoint of CD, then we know:

A) Quadrilateral - 4 sided polygon

1) Kite
2) Parallelogram
a) Rhombus
b) Rectangle
i) Square
3) Trapezoid
a) Isosceles Trapezoid

VI) Areas of Triangles and Quadrilaterals:
A) Rectangles, Square,Parallelogram
A = (base)(height) where the base and height must be perpendicular
B) Triangle
A = (1/2)(base)(height) where the base and height must be perpendicular
C) Trapezoid
A = (1/2)h (B1 + B2)
D) Rhombus and Kite
A =(1/2)( d1 )(d2 )

### Geometry Chapter 6.1 Polygons

6.1 Polygons

check out this website:

http://www.math.com/tables/geometry/polygons.htm

A) Polygon - is a plane figure with 3 or more sides, with each side intersecting with exactly 2 other sides, and that meets the following conditions:
1. If is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear.
2. Each side intersects exactly two other sides, one at each endpoint.
3. each endpoint of a side is a vertex of the polygon.

B) Polygons are named by the numbe of sides they have.
3 sides = triangle
5 sides = pentagon
6 sides = hexagon
7 sides = heptagon
8 sides = octagon
9 sides = nonagon
10 sides = decagon
12 sides = dodecagon
n sides = n-gon
C) A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon.
D) a polygon that that is not convex is called nonconvex or concave.
E) Equilateral - a polygon with all of its side are congruent.
F) Regular - a polygon that is equilateral and equiangular.
G) Diagonal of a Polygon: A segment that joins 2 nonconsecutive vertices.
H) Theorem:
1) Interior Angles of a quadrilateral - the sum of the measures of the interior angles of a quadrilateral is 360 degrees.

## Thursday, July 19, 2007

### Geometry 5.6 Indirect Proof and Inequalities in Two Triangles

Geometry 5.6 Indirect Proof and Inequalities in Two Triangles

I) Theorems:
a) Hinge Theorem - if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
Example: Given ΔRST and ΔVWX, RS = VW and ST = WX, angle S = 100 degrees and angle W = 80 degrees, we can conclude that RT is greater than VX.
B) Converse of the Hinge theorem - if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

II) Using Indirect Proofs:
A) Indirect Proof - is a proof in which you prove that a statement is true by first assuming that its opposite is true. If this assumption leasds to an impossibility, then you have proved that the original statement is true.
1) Guidelines for writing an indirect proof:
a) identify the statement that you want to prove is true.
b) begin by assuming the statement is false; assum its opposite is true.
d) if you obtain a contradiction, then the original statement must be true.

Example: Prove a triangle cannot have 2 right angles.

1) Given ΔABC.
2) Assume angle A and angle B are both right angles is true by one of two possibilities (it is either true or false so we assume it is true).
3) measure of angle A = 90 degrees and measure of angle B = 90 degrees by definition of right angles.
4) measure of angle A + measure of angle B + measure of angle C = 180 degrees by the sum of the angles of a triangle is 180 degrees.
5) 90 + 90 + measure of angle C = 180 by substitution.
6) measure of angle C = 0 degrees by subtraction postulate
7) angle A and angle B are both right angles is false by contradiction (an angle of a triangle cannot equal zero degrees)
8) A triangle cannot have 2 right angles by elimination (we showed since that if they were both right angles, the third angle would be zero degrees and this is a contridiction so therefore our assumption was false ).

Here is another direct proof in t-table form:

Example 2:

This is a direct proof.
Given: LM = MN
Prove: line segment LM is congruent to line segment MN.

This is example 2 as an indirect proof:

Example 3:
Given: LM = MN
Prove: line segment LM is congruent to line segment MN.

Example 4: Given angle PQR is a straight angle, prove the measure of the angle is equal to 180 degrees indirectly.

Example 5: Prove the following indirectly.

As you may have noticed, each indirect proof has 4 steps that have the same concepts:
1. given
2. Assume the opposite of what you want to prove is true.
last step, what you wanted to prove is proven by elimination.

## Wednesday, July 18, 2007

### Geometry Chapter 5.5 Inequalities in One Triangle

Geometry chapter 5.5 Inequalities in One Triangle

I) Theorems:
A) If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
B) If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
C) Exterior Angle Inequality - the measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.
Example: Given ΔABC with side BC extended to point D forming exterior angle ACD,
measure of angle ACD > measure of angle A and
measure of angle ACD > measure of angle B
D) Triangle Inequality - the sum of the lengths of any two sides of a triangle is great than the length of the third side.
Example: Given the triangle has lengths of 5 and 15, what is the other length x?
5 + 15 = 20 and 15 - 5 = 10, therefore the third length is
10 < x < 20

### Geometry Chapter 5.4 Midsegment Theorem

Geometry Chapter 5.4 Midsegment Theorem

I) Vocabulary:
A) Midsegment of a triangle - is a segment that connects the midpoints of two sides of a triangle.
B) Theorem: Midsegment Theorem - the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Example: given ΔABC with point D the midpoint of AC and point E the midpoint of BC and point F is the midpoint of AB, we can conclude
1) DE ll AB
2) DE = 1/2 AB
3) EF ll AC
4) EF = 1/2 AC
5) FD ll BC
6) FD = 1/2 BC
Therefore, you end up with 4 triangles that are congruent.
Check out websites dealing with Fractals- a fractal is created with midsegments. Beginning with any triangle, shade the triangle formed by the three midsegments. Continue this process for each unshaded triangle. Here is one:
http://mathforum.org/alejandre/applet.mandlebrot.html

### Geometry Chapter 5.3 Medians and Altitudes of a Triangle

Geometry Chapter 5.3 Medians and Altitudes of a Triangle

I) Vocabulary:
A) Median of a triangle - is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
B) Centroid of the triangle - the point of concurrency of the three medians of a triangle .
C) Altitude of the triangle - is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle.
D) Orthocenter of the triangle - the point of concurrency of the three altitudes of a triangle.

II) Theorems:
A) Concurrency of Medians of a Triangle - the medians of a triangle intersect at a point that is called the centroid and that is two thirds of the distance from each vertex to the midpoint of the opposite side.
Example: If point P is the centroid of ΔABC, then AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3 CE.
B) Concurrency of Altitudes of a Triangle - the lines containing the altitudes of a triangle are concurrent at the orthocenter.
Example: If AE, BF, and CD are the altitudes of ΔABC, then the lines AE, BF and CD intersect at the orthocenter point H.

### Geometry Chapter 5.2 Bisectors of a Triangle

Geometry Chapter 5.2 Bisectors of a Triangle

I) Vocabulary:
A) A Perpendicular bisector of a triangle
- is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.
B) Concurrent lines - when three or more lines (or rays or segments) intersect in the same point.
C) Point of concurrency - the point of intersection of the concurrent lines.
D) Circumcenter of the triangle - the point of concurrency of the perpendicular bisectors of a triangle.
E) Angle bisector of a triangle - is a bisector of an angle of the triangle.
F) Incenter of the triangle - the point of concurrency of the angle bisectors of a triangle.
II) Theorems:
A) concurrency of Perpendicular Bisectors of a Triangle - the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
Example: Given ΔABC with point P being the circumcenter of the triangle, we can conclude that
PA = PB = PC.
Using point P as the center of the circle and PA, PB, or PC as a radius, the circleis circumscribed about the triangle.
B) Concurrency of Angle Bisectors of a Triangle - the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.
Example: Given ΔABC with point P being the incenter of the triangle, we can conclude that
PD = PE = PF and PD ll BC, PE ll AC and PF ll AB.
Using point P as the center of the circle and PD, PE, or PF as a radius, the circle is inscribed within the triangle.

### Geometry Chapter 5.1 Perpendiculars and Bisectors

Geometry Chapter 5.1 Perpendiculars and Bisectors:

I) Vocabulary:
A) Perpendicular Bisectors - a segment, ray, line or plane that is perpendicular to a segment at its midpoint.
B) Equidistant - a point is equidistant from two points if its distance from each point is the same.
C) Distance from a point to a line - is defined as the length of the perpendicular segment from the point to the line.
D) Equidistant from the two lines - when a point is the same distance from one line as it is from another line.
II) Theorems:
A) Perpendicular Bisector Theorem - if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Example: If line CP is the perpendicular bisector of line segment AB, then CA = CB.
B) Converse of the Perpendicular Bisector Theorem - if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Example: If DA = DB, then point D lies on the perpendicular bisector of line segment AB.
C) Angle Bisector Theorem - if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Example: if measure of angle BAD = measure of angle CAD, then BD = DC.
D) Converse of the Angle Bisector theorem - if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.
Example: If DB = DC, then measure of angle BAD = measure of angle CAD

## Tuesday, July 17, 2007

### Geometry Chapter 4.7 Triangles and Coordinate Proof

Geometry Chapter 4.7 Triangles and Coordinate Proof

A coordinate proof involves placing geometric figures in a coordinate plane. Then you will use:

1) Distance Formula - to show segments are congruent
2) Midpoint formula - to show that the segment is bisected
3) Slope formula -
a) if 2 lines have the same slope, then the lines are parallel
b) if 2 lines have slopes that are negative reciprocals of each other, then the lines are perpendicular

Example: To show that ΔABC is an isosceles right triangle given A(0,0), B(0,6) and C(6,0).
1. slope of AB = (0-6)/(0-0) = undefined
2. slope of BC = (6 - 0)/(0 - 6) = 6/(-6) = -1
3. slope of AC = (0 - 0)/(0 - 6) = 0 / 6 = 0

Therefore AB is perpendicular to AC because horizontal and vertical lines are perpendicular to each other. therefore angle A is a right angle because perpendicular lines form right angles.

4. AB = 6, AC = 6, and BC = 6√2
so we can conclude that AB = AC because they have the same measure.

Therefore since ΔABC has one right angle and 2 congruent sides, it is an isosceles right triangle.

### Geometry Chapter 4.6 Isosceles, Equilateral, and Right Triangles

Geometry Chapter 4.6 Isosceles, Equilateral, and Right Triangles

I) Vocabulary:
A) Base angles - the two angles adjacent to the base of an isosceles triangle.
B) Vertex angles - the angle opposite the base of an isosceles triangle.
II) Theorems:
A) Base Angles Theorem: if two sides of a triangle are congruent, then the angles opposite them are congruent.
B) Converse of the Base Angles Theorem: if two angles of a triangle are congruent, the the sides opposite them are congruent.
C) Hypotenuse - Leg Congruence Theorem (HL = HL): if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.
Example: Given Δ ABC and ΔDEF are both right triangles, BC = EF and AC = DF,
then ΔABC = ΔDEF.
III) Corollaries:
A) If a triangle is equilateral, then it is equiangular.
B) If a triangle is equiangular, then it is equilateral.

### Geometry Chapter 4.3 and 4.4 Triangle are Congruent: SSS, SAS, ASA, and AAS

Geometry Chapter 4.3 Triangle are Congruent by SSS and SAS

I) Postulates:
SSS = SSS Congruence Postulate - if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if AB = DE, BC = EF and AC = DF, then ΔABC = ΔDEF.

SAS = SAS Congruence Postulate - if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if AB = DE, BC = EF and ∠B = ∠E, then ΔABC = ΔDEF.

Geometry Chapter 4.4 Triangles are Congruent by ASA and AAS

II) Postulates:
ASA = ASA Congruence Postulate - if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if angle A = angle D, AB = DE, and angle B = angle E, then Δ ABC = Δ DEF.

AAS = AAS Congruence Postulate - if two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent.
Example: Given Δ ABC and Δ DEF,
if angle A = angle D, angle C = angle F, and BC = EF , then Δ ABC = Δ DEF.

### Geometry chapter 4.2 Congruence and Triangles

Geometry Chapter 4.2 Congruence and Triangles

I) Vocabulary:
A) When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.

Example: Given Δ ABC is congruent to Δ PQR then we know
1) angle A = angle P, angle B = angle Q, and angle C = angle R
2) AB = PQ, BC = QR, and AC = PR

Make sure that you list the corresponding angles in the same order with the triangle congruence.
Example: ΔABC = ΔDEF is not the same as ΔABC = ΔEFD.

B) Third Angles Theorem - if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
C) Reflexive Postulate - Every triangle is congruent to itself
D) Symmetric Postulate - If ΔABC = ΔDEF, then ΔDEF = ΔABC
E) Transitive Postulate - If ΔABC = ΔDEF and ΔDEF = ΔJKL, then ΔABC = ΔJKL

### Geometry Chapter 4.1 Triangles and Angles

Geometry Chapter 4.1 Triangles and Angles

I) Vocabulary:
A) Triangle - is a figure formed by three segments joining three noncollinear points.
B) Classification of triangles:
1) By sides:
a) Equilateral Triangle - has 3 congruent sides
b) Isosceles Triangle - has at least 2 congruent sides
c) Scalene Triangle - has no congruent sides
2) By angles:
a) Acute Triangle - has 3 acute angles
b) Equiangular Triangle - has 3 congruent angles that measure 60 degrees each
c) Right Triangle - has one right angle and 2 acute angles
d) Obtuse Triangle - has one obtuse angle and 2 acute angles
C) Vertex - each of the three points joining the sides of a triangle (plural - vertices)
D) Adjacent Sides - in a triangle, two sides sharing a common vertex.
E) Right triangles have 2 sides that form the right angle called the legs. The side opposite the right angle is the hypotenuse of the triangle.
F) Interior angles - when the sides of a triangle are extended, other angles are formed. the three original angles are the interior angles.
G) Exterior angles - the angles that are adjacent to the interior angles.
H) Theorem:
1) Triangle sum theorem - the sum of the measures of the interior angles of a triangle is 180 degrees.
2) Exterior Angle Theorem - the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
I) Corollary to a theorem - is a statement that can be proved easily using the theorem.
Example: The acute angles of a right triangle are complementary.

Example: given Triangle ABC with side BC extended through point D, if angle A = 65 degrees and angle ACD = 2x + 10 and angle B = x, solve for x.

angle A + angle B = angle ACD
65 + x = 2x + 10
55 = x

## Monday, July 16, 2007

### Geometry Chapter 3.7 Perpendicular lines in the Coordinate Plane

Geometry chapter 3.7 Perpendicular lines in the Coordinate Plane:

I) Postulate:
A) In a coordinate plane, 2 non-vertical lines are perpendicular if and only if they product of their slopes is equal to (-1). The two slopes are negative reciprocals of each other.

all vertical and horizontal lines are perpendicular to each other.

Given the slope of the first line is a/b,
then the slope of the perpendicular line is (-b/a).

(a/b)(-b/a) = (-1)

B) To show that two lines are perpendicular lines:
1) Find the slopes of each line
2) multiply the slopes
3) Perpendicular if they = -1

Example: Given the first line has points (-1,2) and (5, -1) and the second line has points (4,2) and (1, -4), find out if the two lines are perpendicular:

M1 = (2- (-1))/(-1 - 5) = 3/(-6) = -1/2
M2 = (2 - (-4))/(4 - 1) = 6/3 = 2

when you multiply these together (-1/2)(2) = -1
so these lines are perpendicular to each other.

Example 2:
Given the slope of the first line is 3/4, what is the slope of a line perpendicular to this line and what is the equation of the line perpendicular through the point (9, -2)

y = mx + b
the slope was 3/4 so the perpendicular slope is (-4/3)

-2 = (-4/3)(9) + b
-2 = -12 + b
10 = b

therefore the equation of the line perpendicular to the first line through the point (6, -2) is
y = (-4/3) x + 10

### Geometry Chapter 3.6 Parallel lines in the Coordinate Plane

Geometry Chapter 3.6 Parallel lines in the Coordinate Plane

I) Vocabulary:
A) Slope: (nonvertical line): Ratio of the vertical change (the rise, Δ y) to the horizontal change (the run, Δ x)

Formula: slope = m = (Δy/Δx) = (y2 - y1)÷ (x2 - x2)

II) Postulate:
A) In the coordinate plane, 2 nonvertical lines are parallel if and only if they have the same slope. Any two vertical or horizontal lines are parallel.
B) Equation of a line:
y = mx + b
where m is the slope of the line and b is the y-intersept (where it crosses the y-axis)

### Geometry 3.3 Parallel Lines and Transversals and 3.4 Proving lines are parallel and 3.5 Using Properties of Parallel lines

Geometry 3.3 Parallel Lines and Transversals

I) Vocabulary:
A) Transversal: A line that intersects 2 lines at different places.
II) Postulate:
A) If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
B) If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
C) If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
D) If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
E) If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

II) Converse of above:
A) If 2 lines are cut by a transversal such that the corresponding angles are congruent, then the lines are parallel.
B) if 2 lines are cut by a transversal such that the pairs of consecutive interior angles are supplementary, then the lines are parallel.
C) if 2 lines are cut by a transversal such that the pairs of alternate exterior angles are congruent, then the lines are parallel.
D) if 2 lines are cut by a transversal such that the pairs of alternate interior angles are congruent, then the lines are parallel.

III) Using Properties of Parallel lines:
A) if two lines are parallel to the same line, then they are parallel to each other.
B) In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

### Geometry 3.2 Proof and Perpendicular lines

Geometry 3.2 Proof and Perpendicular Lines:

I) Theorems:
1. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
2. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
3. If two lines are perpendicular, then they intersect to form four right angles.

### Geometry Chapter 3.1 Lines and angles

Geometry Chapter 3.1 Lines and Angles:

I) Vocabulary:
A) Parallel Lines (ll): two lines that are coplanar and do not intersect.
B) Skew lines: two lines that doe not intersect and are not coplanar.
C) Parallel Planes: Planes that do not intersect.
II) Postulates:
A) If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
B) If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.
III) Identifying Angels formed by transversals:
A) Transversal: is a line that intersects two or more coplanar lines at different points.
B) Corresponding angles: if you have two lines that are intersected by a transversal, the the two angles that occupy the same position - (eg. to the left of the transversal and on top of the two lines)
C) Alternate interior angles: if you have two lines that are intersected by a transversal, then the two angles that lie between the two lines and on opposite sides of the transversal.
D) Alternate exterior angles: if you have two lines that are intersected by a transversal, then the two angles that lie outside the two lines on opposite sides of the transversal.
E) Consecutive Interior Angles: two angles if they lie between the two lines on the same side of the transversal. Also known as same side interior angles.

### Geometry 2.5 Proving Statements about Segments and 2.5 Proving Statements about Angles

Geometry 2.5 Proving Statements about Segments:

I) Vocabulary:
A) Reflexive: any segment or angle is congruent to itself.
B) Symmetric: If AB = CD, then CD = AB or angle A = angle B, then angle B = angle A.
C) Transitivity: If AB = CD and CD = EF, then AB = EF or if angle A = angle B and Angle B = angle C, then angle A = angle C.
D) Theorem: a true statement that follows as a result of other true statements.
All theorems must be proven true for all cases. Here are a few ways of doing theorems.

1. Two-Column Proof: has numbered statements and reasons that show the logivcal order of an argument.
2. Paragraph proof: a proof can be written in paragraph form.
3. Flow proof: a chart that has arrows going from one statement to the next with the reasons written underneath the statement.

E) Theorems:
1. All right angles are congruent.
2. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
3. If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
4. If two angles form a linear pair, then they are supplementary.
5. Vertical angles are congruent.
6. Corresponding parts of congruent triangles are congruent (CPCTC)
7. If two lines form congruent adjacent angles, then the lines are perpendicular.
8. The supplement of a right angle is a right angle.

### Geometry Chapter 2.3 Deductive Reasoning and 2.4 Reasoning with Properties from Algebra

Geometry Chapter 2.3 Deductive Reasoning

I) Using Symbolic Notation:
1) " → " means implies
2) "~" means not
3) "↔" means if and only if

A) conditional statement has a hypothesis (symbolically "p") and a conclusion (symbolically "q") so: if p then q or p → q
B) Converse: q → p
C) Inverse: ~p → ~ q
D) Contrapositive: ~q → ~ p
E) Biconditional: p ↔ q

F) Deductive Reasoning: uses facts, definitions, and accepted properties in a logical order to write a logical argument.
1) Law of detachment: if "p → q" is a true statement and "p" is a true statement, then we can conclude that "q" is true.
Example: If Mark gets time off of work then he is going to Hawaii on vacation. (this is true)
Mark gets time off of work. (this is true).
Therefore we can conclude that Mark is going to Hawaii on vacation.

Symbolically:
Let "p" be Mark gets time off of work.
Let "q" be Mark is going to Hawaii on vacation.

p → q is true
p is true.
Therefore q is true.

2) Law of Syllogism:
If p → q AND q → r are both true, then we can conclude that p → r .
This is like the transitivity postulate.

Example: If Carol buys a new swimsuit, then she is going to the Lake.
If Carol goes to the Lake, then she will go swimming.
Therefore we can conclude that

2.4 Reasoning with Properties from Algebra:

G) Postulates:
1. Reflexive Postulate: a = a
2. Symmetric Postulate: if a = b then b = a
3. Transitivity Postulate: if a = b and b = c, then a = c
4. Addition Postulate: if a = b and c is not equal to 0, then a + c = b + c
5. Subtraction Postulate: if a = b and c is not equal to 0, then a - c = b - c
6. Multiplication Postulate: if a = b and c is not equal to 0, then ac = bc
7. Division Postulate: if a = b and c is not equal to 0, then a/c = b/c
8. Distribution Postulate: if a(b + c) then ab + ac

H) Definitions:
1. Perpendicular lines - form right angles
2. Right angle - an angle that measures 90 degrees.

### Geometry Chapter 2.2 Definitions and Biconditional Statements

Geometry Chapter 2.2 Definitions and Biconditional Statements

I) Vocabulary:
A) Perpendicular: Two lines that intersect to form right angles.
B) Line Perpendicular to a Plane: A line that intersects a plane in a point and is perpendicular to every line that includes that point in the plane that intersects it.
C) Biconditional Statement: A statement that contains the words "if and only if" (iff) and is equivalent to writing a statement combining a conditional statement and its converse. For the truth value of a biconditional statement to be true, both the conditional statemene and its converse have to have the same truth value.

Example: Conditional Statement: If two sides of a triangle are congruent, then the angles opposite them are congruent.
Converse Statement: If two angles of a triangle are congruent, then the sides opposite them are congruent.

Biconditional Statement: Two sides of a triangle are congruent if and only if the two angles of the triangle are congruent.

### Geometry Chapter 2.1 Conditional Statements

Geometry chapter 2.1 Conditional Statements:

I) Vocabulary:

A. Conditional Statements - (if-then) - has two parts:

1. Hypothesis - (if)

2. Conclusion - (then)

Example: If it is noon in Georgia, then it is 9 am in California.

B) Converse Statement: Switch the hypothesis and conclusion of a Conditional Statement.

Example: If it is 9 am in California, then it is noon in Georgia.

C) Inverse Statement: negate both the hypothesis and conclusion of a Conditional Statement.

Example: If it is not noon in Georgia, then it is not 9 am in California.

1) Negation - write the negative of the statement.
Example: Statement: Angle A is acute.
Negation: Angle A is not acute.
or
It is not true that angle A is acute.
D) Contrapositive Statement: Switch and negate both the Hypothesis and Conclusion of a Conditional Statement.

Example: If it is not 9 am in California, then it is not noon in Georgia.

E) Equivalent Statements: 2 statements that are both true or both false, they have the same truth value. Contrapositive statements is always equivalent to its Conditional statement. The converse statement is always equivalent to its inverse statement.

F) Counterexample: An example that shows that a conditional statement is false.

Example: If x2 = 25, then x = 5
A counterexample is x = (-5) because (-5)2 = 25 but 5 is not equal to (-5)

II) Postulates:

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

## Thursday, July 12, 2007

### Geometry Chapter 1.7 Introduction to Perimeter, Circumference, and Area

Geometry Chapter 1.7 Introduction to Perimeter, Circumference, and Area

I) Vocabulary and Formulas:

Notation:
A = Area
P = Perimeter

A) Triangle:
A = (1/2)(base)(height)
P = (side a) + (side b) + (side c)

Example: Given Triangle ABC with side a = 10, b = 17 and c = 21 and the height of the triangle is 8. Find the area and the perimeter.

Area = (1/2)(21)(8) = 84 square units.
Perimeter = 10 + 17 + 21 = 48 units

B) Square: has all the sides equal so the base = height so we will call them all sides.
A = (side)(side) = s2
P = side + side + side + side = 4s

Example: let one side of a square is equal to 5 inches. what is the area and perimeter?
A = (5)2 = 25 square inches
P = 4 (5) = 20 inches

C) Rectangle:
A = (base)(height)
P = 2b + 2h

Example: let the base = 3 and the height = 4
A = (3)(4) = 12 square units
P = (2)(3) + (2)(4) = 6 + 8 = 14 units

D) Circle: 2 radius = diameter
A = π r2
P = Circumference = 2 π r

Example: If a circles diameter is 20 cm, what is the area and circumference?
A = π (10)2 = 100π square cm.
C = (2) π (10) = 20 π cm.

### Geometry Chapter 1.6 Angle Pair Relationships

Geometry Chapter 1.6 Angle Pair Relationships

I) Vocabulary

A) Vertical Angles: is when the sides of 2 angles form 2 pairs of opposite rays.
1. Vertical angles are congruent.

B) Linear Pair: 2 adjacent angles are a linear pair if their noncommon sides are opposite rays.
1. If two angles form a linear pair, then they are supplementary.
2. If two angles form a linear pair, then their sum is 180 degrees.

C) Complementary Angles: 2 angles whose measures total 90 degrees.
1. If two angles have the same complement, then the 2 angles are congruent.
2. If two angles are complementary to congruent angles, then the two angles are congruent.

D) Supplementary Angles: 2 angles whose measures total 180 degrees.
1. If two angles have the same supplement, then the 2 angles are supplementary.
2. If two angles are supplementary to congruent angles, then the two angles are supplementary.

### Geometry Chapter 1.5 Segment and Angle Bisectors

I) Vocabulary:

A) Midpoint: A midpoint of a segment is the point that divides, or bisects, the segment into two congruent segments.

B) Segment Bisector: is a segment, ray, line, or plane that intersects a segment at its midpoint.

Example: Given segment AB bisects segment CD at point E. If CD = 10, what does AE and EB equal?
10/2 = 5 so they both equal 5 units.

C) Angle Bisector: is a ray that divides an angle into two adjacent angles that are congruent.

Example: Given angle PQR is bisected by ray QS, and measure of angle SQR = 22 degrees, what is the measure of angle PQR and angle PQS?

22 x 2 = 44 so measure of angle PQR = 44 degrees and when an angle is bisected, the two angles formed are congruent so angle SQR = angle PQS = 22 degrees.

D) Midpoint Formula: the mean, or average, of the x-coordinates and the y-coordinates.

therefore the midpoint formula is: ((x1+ x2) /2, (y1+ y2) /2)
example: Given A(-1, 7) and B(3, -3) what is their midpoint?

((-1 + 3)/2, (7 + -3)/2) = (2/2, 4/2) = (1, 2)

### Geometry chapter 1.4 Angles and their measures

Geometry Chapter 1.4 Angles and their Measures:
I) Vocabulary and Postulates:

A) Angle - consists of two different rays that have the same initial point.
1) Vertex - the initial point of the angle.
2) Rays of an angle - the sides of the angle.

B) Congruent Angles - angles that have the same measure.
C) Angle Addition Postulate - If P is on the interior of angle RST, then measure of angle RSP + measure of angle PSP = measure of angle RST.

D) Acute Angle - an angle that's measure is between zero and ninety degrees.

E) Right Angle - an angle that's measure is 90 degrees.

F) Obtuse Angle - an angle that's measure is between 90 degrees and 180 degrees.

G) Straight Angle - an angle that's measure is 180 degrees.

H) Adjacent Angles - two angles are adjacent angles if they share a common vertex AND a common side, but do not have any common interior points.

## Tuesday, July 10, 2007

### Geometry Chapter 1.3 Segments and Their Measures

Geometry Chapter 1.3 Segments and Their Measures

I) Vocabulary & Postulates:

A) theorems: Rules that have been proven to be true.
Example: Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180 degrees.

B) Postulates or Axioms: Rules that we accept as true without proof.

C) Between - When 3 points are one a line, you may say that 1 point is between the others.

D) Segment Addition Postulate or Partition Postulate: If B is between point A and point C, then
AB + BC = AC

E) Distance formula: AB = √((x1 - x2)2 + (y1 - y2)2)

this is derived from the pythagorean theorem, where the difference of the x's is represented by "a" and the difference of the y's is represented by "b" and AB = c so

a2 + b2 = c2

F) Congruent Segments - Segments that have the same measure

Written:

Lengths are equal in length so: AB = CD

and

Segments are congruent so: line segment AB is congruent to line segment CD.

### Geometry Chapter 1.2 Points, Lines and Planes

Chapter 1.2 Points, Lines, and Planes

I) Undefined Terms:

A) Point: a point has no dimension, usually represented by a small dot.

B) Line: A line extends in one dimension. It is usally represented by a straight line with two arrowheads to indicate that the line extends without end in two directions.

C) Plane: a plane extends in two dimensions, goes on forever.

D) Collinear Points: 2 or more points on the same line.

E) Coplanar Points: Points that lie on the same plane.

F) Line Segment: a segment of a line; consists of 2 endpoints and all points in between.

G) Ray: An initial point and all points one side of that point.

H) Opposite Rays: 2 rays that form a straight line.

I) Intersect: Two or more geometric figures intersect if they have one or more points in common.

### Geometry - Chapter 1.1 Patterns and Inductive Reasoning

Chapter 1.1 - Patterns and Inductive Reasoning

A. Using Inductive Reasoning
1. Look for a pattern
. - look at several examples. Use diagrams and tables to help discover a pattern.
2. Make a Conjecture. - a conjecture is an unproven statement that is based on observation.
3. Verify the Conjecture - use logical reasoning to verify that the conjecture is true in all cases.

Example - the sum of the first n odd positive integers is...
first odd positive integer is 1 = 1
sum of first two odd positive integers is 1 + 3 = 4
sum of first three odd positive integers is 1 + 3 + 5 = 9
sum of first four odd positive integers is 1 + 3 +5 + 7 = 16

looking at these answers: 1, 4, 9, 16 we see they are perfect squares so
12, 22, 32, 42
Therefore we can make a conjecture that the sum of the first n odd positive integers is n2 .

B. Counterexample - is an example that shows a conjecture is false.
Example: All prime numbers are odd.
Since 2 is a prime number but 2 is even this would be a counterexample.