Friday, September 29, 2006
Notes 3-6
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)
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)
Thursday, September 28, 2006
Wednesday, September 27, 2006
Notes 3-5
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
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
Tuesday, September 26, 2006
Notes 3-4
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
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
Monday, September 25, 2006
Notes 3-2 and 3-3
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
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
Notes: 3.1
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
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
Thursday, September 21, 2006
Wednesday, September 20, 2006
Tuesday, September 19, 2006
Monday, September 18, 2006
Notes 2-6
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
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
HW: from assignment sheet: all problems listed for page 112, all problems EXCEPT 23 and 24 for page 114
Friday, September 15, 2006
HW:9-15
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
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!
SORRY THAT I WAS LATE POSTING TODAY! HAD A FEW ERRANDS TO RUN RIGHT AFTER SCHOOL! SEE YOU AT THE FOOTBALL GAME!
Thursday, September 14, 2006
Notes: 2-3 and 2-4 *Not all symbols appear correctly in notes
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 pq 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 pq is a true conditional statement and p is true, then q is true
(ex 4 on pg 89)
Law of Syllogism: if pq and qr 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
-Geometric: If AB=CD, then CD=AB;;;;; If m
-Geo: If AB=CD and CD=ED, then AB=EF;;;; If m
Example 1 and 2 pg 96; 4 and 5 pg 98
Guided practice 1-9 pg 99
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 pq 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 pq is a true conditional statement and p is true, then q is true
(ex 4 on pg 89)
Law of Syllogism: if pq and qr 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
-Geometric: If AB=CD, then CD=AB;;;;; If m
-Geo: If AB=CD and CD=ED, then AB=EF;;;; If m
Example 1 and 2 pg 96; 4 and 5 pg 98
Guided practice 1-9 pg 99
Wednesday, September 13, 2006
Notes end of 2-1, also 2-2
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
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
Tuesday, September 12, 2006
Notes 2-1 part 1
2-1 Conditional Statements
Conditional Statement- contains a hypothesis and a conclusion. Generally, they are written in “if-then” format where the “if” is the hypothesis and the “then” os the conclusion.
If I get an A on the final, then I will pass this class.
Hypoth Concl.
Rewrite into if-thens:
1) Two points are collinear if they lie on the same plane
2) All sharks have a boneless skeleton
3) A # divisible by 9 is also divisible by 3
Converse- of a conditional statement is made by switching the hypoth and conclusion
Ex. Statement: If I am home by 8, then I can watch House tonight.
Converse: If I can watch House tonight, then I am home by 8.
Try: If two segments are congruent, then they have the same length.
Ans: If two segments have the same length, then they are congruent
Negation- the negative of the statement
Ex. m∟A=30, angle A is acute
Negation: m angle A≠30, angle A is not acute
Inverse- formed when the hypoth and conclusion of a conditional statement are negated
Contrapositive- formed when the hypoth and conclusion of a converse statement are negated.
Original If m angle A=30, then angle A is acute
Inverse If m angle A≠30, angle A is not acute
Converse If angle A is acute, then m angle A=30
Contrapositive If angle A is not acute, then m angle A≠30
Conditional Statement- contains a hypothesis and a conclusion. Generally, they are written in “if-then” format where the “if” is the hypothesis and the “then” os the conclusion.
If I get an A on the final, then I will pass this class.
Hypoth Concl.
Rewrite into if-thens:
1) Two points are collinear if they lie on the same plane
2) All sharks have a boneless skeleton
3) A # divisible by 9 is also divisible by 3
Converse- of a conditional statement is made by switching the hypoth and conclusion
Ex. Statement: If I am home by 8, then I can watch House tonight.
Converse: If I can watch House tonight, then I am home by 8.
Try: If two segments are congruent, then they have the same length.
Ans: If two segments have the same length, then they are congruent
Negation- the negative of the statement
Ex. m∟A=30, angle A is acute
Negation: m angle A≠30, angle A is not acute
Inverse- formed when the hypoth and conclusion of a conditional statement are negated
Contrapositive- formed when the hypoth and conclusion of a converse statement are negated.
Original If m angle A=30, then angle A is acute
Inverse If m angle A≠30, angle A is not acute
Converse If angle A is acute, then m angle A=30
Contrapositive If angle A is not acute, then m angle A≠30
Monday, September 11, 2006
HW: 9/11- None!
Had the first chapter test today, so there is no homework. It is a well deserved break! Hopefully I will be in tomorrow and we can go over the test.
There was the extra credit assignment that you may be doing for homeowrk, also.
There was the extra credit assignment that you may be doing for homeowrk, also.
Thursday, September 07, 2006
Notes: End of 1-5; 1-6 and 1-7
Bisecting an angle:
An angle bisector: is a ray that divides and angle into two adjacent angles that are congruent (adjacent- share one common side and vertex but no common interior points)
Draw angle bisector (pg 36)à check with protractor
Work through ex. 3-5 on pg 37
Geometry angle worksheet (also have them draw in the angle bisectors)
1-6 Angle Pair Relationships:
vertical angles- have sides that form 2 pairs of opposite rays (opposite angles will be congruent)
linear pairs- adjacent angles whose non-common sides are opposite rays (the measures of linear pairs must add up to 180
complementary angles have measurements that add up to 90 degrees (can be adjacent or not)
supplementary angles: have measurements that add up to 180 degrees (can be adjacent or not)
practice ws 1.6
1.7 Intro to Perimeter, Circumference, and Area *********Note Blogger does not always properly display "pi" or exponents*************
Square: P=4s
A=s2 (side squared)
Rectangle: P=2l+2w
A=lw
Triangle:P=a+b+c
A= ½ bh
Circle: radius, r
Circumference( distance around the edge of circle)= C=2 r (2 *pi* r)
A= r2 (pi*(r squared))
An angle bisector: is a ray that divides and angle into two adjacent angles that are congruent (adjacent- share one common side and vertex but no common interior points)
Draw angle bisector (pg 36)à check with protractor
Work through ex. 3-5 on pg 37
Geometry angle worksheet (also have them draw in the angle bisectors)
1-6 Angle Pair Relationships:
vertical angles- have sides that form 2 pairs of opposite rays (opposite angles will be congruent)
linear pairs- adjacent angles whose non-common sides are opposite rays (the measures of linear pairs must add up to 180
complementary angles have measurements that add up to 90 degrees (can be adjacent or not)
supplementary angles: have measurements that add up to 180 degrees (can be adjacent or not)
practice ws 1.6
1.7 Intro to Perimeter, Circumference, and Area *********Note Blogger does not always properly display "pi" or exponents*************
Square: P=4s
A=s2 (side squared)
Rectangle: P=2l+2w
A=lw
Triangle:P=a+b+c
A= ½ bh
Circle: radius, r
Circumference( distance around the edge of circle)= C=2 r (2 *pi* r)
A= r2 (pi*(r squared))
HW: 9/7-9/11
PG 47: 8-12 EVEN, 14-19, 20-26 EVEN, 28-36
Pg 55; 9-20, 23, 25, 31, 35, 36, 44, 46
Due Monday: pg 63 chapter test; pg 64 chapter standardized test
Exam on Monday Ch1
Pg 55; 9-20, 23, 25, 31, 35, 36, 44, 46
Due Monday: pg 63 chapter test; pg 64 chapter standardized test
Exam on Monday Ch1
Wednesday, September 06, 2006
Notes: end of 1.4, also 1.5
Angle addition postulate pg 27:
If P is in the interior of ∟RST (meaning it is between points that lie on each side of the triangle) then m∟RSP + m∟PST= m∟RST
Acute angles 0º< m∟A<90º
Right angles m∟A=90º
Obtuse angles 90º< m∟A<180º
Straight angle m∟A=180º
1.5: Segment and Angle Bisectors
Midpoint of a segment divides or bisects the segment into two congruent segments.
Segment bisector- segment, ray, line, or plane that intersects a segment at the midpoint.
*If two lines or angles are congruent, we use congruence marks to indicate (same # of tick marks)
A construction geometric drawing that uses limited set of tools
Draw segment bisector and midpoint
Midpoint Formula: If A (x1, y1) and B(x2,y2) are points in a coordinate plane, then the midpoint of line segment AB has coordinates:
EX: Find the midpoint of segment AB with the endpoints A(-2,3) and B(5, -2)
EX: the midpoint of line segment RP is M(2,4) one endpoint is R(-1, 7) what is the other endpoint?
If P is in the interior of ∟RST (meaning it is between points that lie on each side of the triangle) then m∟RSP + m∟PST= m∟RST
Acute angles 0º< m∟A<90º
Right angles m∟A=90º
Obtuse angles 90º< m∟A<180º
Straight angle m∟A=180º
1.5: Segment and Angle Bisectors
Midpoint of a segment divides or bisects the segment into two congruent segments.
Segment bisector- segment, ray, line, or plane that intersects a segment at the midpoint.
*If two lines or angles are congruent, we use congruence marks to indicate (same # of tick marks)
A construction geometric drawing that uses limited set of tools
Draw segment bisector and midpoint
Midpoint Formula: If A (x1, y1) and B(x2,y2) are points in a coordinate plane, then the midpoint of line segment AB has coordinates:
EX: Find the midpoint of segment AB with the endpoints A(-2,3) and B(5, -2)
EX: the midpoint of line segment RP is M(2,4) one endpoint is R(-1, 7) what is the other endpoint?
Tuesday, September 05, 2006
hw: 9/5
HW: PG 21: 20, 24-32 EVEN 34, 35, 38, 41, 46, 47
pg 29: 19, 20, 22, 26, 28, 29-37, 38, 41, 43, 55-59
pg 29: 19, 20, 22, 26, 28, 29-37, 38, 41, 43, 55-59
Notes 1-3 and 1-4
1-3 Segments and Their Measures:
Postulate, or Axiom- rules accepted without proof.
Theorem- rules that are proved
Postulate 1-Ruler Postulate: Points on a line can be matched 1 to 1 with the real numbers. These real numbers are the coordinate of the point.
The distance between points A and B, written AB, is the absolute value of the difference between the coordinates of A and B.
AB is also the length of ĀB
In order for one point to be between two others, all three must be collinear. (EX)
Postulate 2- Segment Addition Postulate:
If B is between C and A, then AB+BC=AC
If AB+ BC=AC, then B is between A and C
Google map worksheet one side/ Activity lesson opener 1-3 the other side
Distance Formula- used for computing the distance between two points in a coordinate plane
Length of AB= which is just rearranging the Pythagorean theorem (a2+b2=c2)
Plot the points (1,4) and (10, 8) and find the distance using the formula
Try (-9,14) and (5, -10)
Segments that have the same length are called congruent
1-4 Angles and their Measures
angle- two rays with the same initial point, or vertex. The rays make the sides of the angle
-To name an angle: use 3 letters, the vertex MUST be in the middle and the two others on different sides of the angle. You can also use just the name of the vertex if it cannot be confused with multiple angles. ∟ CAB, ∟A
- To measure an angle, use a protractor which uses degrees.
- m∟ABC= 50º
- If two angles have the same measurement, they are congruent
Angle addition postulate pg 27:
If P is in the interior of ∟RST (meaning it is between points that lie on each side of the triangle) then m∟RSP + m∟PST= m∟RST
Acute angles 0º< m∟A<90º
Right angles m∟A=90º
Obtuse angles 90º< m∟A<180º
Straight angle m∟A=180º
Postulate, or Axiom- rules accepted without proof.
Theorem- rules that are proved
Postulate 1-Ruler Postulate: Points on a line can be matched 1 to 1 with the real numbers. These real numbers are the coordinate of the point.
The distance between points A and B, written AB, is the absolute value of the difference between the coordinates of A and B.
AB is also the length of ĀB
In order for one point to be between two others, all three must be collinear. (EX)
Postulate 2- Segment Addition Postulate:
If B is between C and A, then AB+BC=AC
If AB+ BC=AC, then B is between A and C
Google map worksheet one side/ Activity lesson opener 1-3 the other side
Distance Formula- used for computing the distance between two points in a coordinate plane
Length of AB= which is just rearranging the Pythagorean theorem (a2+b2=c2)
Plot the points (1,4) and (10, 8) and find the distance using the formula
Try (-9,14) and (5, -10)
Segments that have the same length are called congruent
1-4 Angles and their Measures
angle- two rays with the same initial point, or vertex. The rays make the sides of the angle
-To name an angle: use 3 letters, the vertex MUST be in the middle and the two others on different sides of the angle. You can also use just the name of the vertex if it cannot be confused with multiple angles. ∟ CAB, ∟A
- To measure an angle, use a protractor which uses degrees.
- m∟ABC= 50º
- If two angles have the same measurement, they are congruent
Angle addition postulate pg 27:
If P is in the interior of ∟RST (meaning it is between points that lie on each side of the triangle) then m∟RSP + m∟PST= m∟RST
Acute angles 0º< m∟A<90º
Right angles m∟A=90º
Obtuse angles 90º< m∟A<180º
Straight angle m∟A=180º
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