Angle Addition Postulate
-GOALSTo state what the angle
addition postulate is
Show examples of how to
solve a problem with the
angle addition postulate
Rules- A Better
Understanding
• The angle addition postulate basically states
that if point M is in the interior of ∠JKL, then,
∠JKM + ∠LKM = ∠JKL.
• The rule is that if an angle is split by a
bisector, the two mini angles now are
equivalent to the whole angle .
Practice Problems
Quick Time™a nd a
dec ompr esso r
ar e nee ded to see this pictur e.
• If m∠ABC=72˚ and m∠CBD=38˚, what is m∠ABD?
*ANSWER: ∠ABD is 110˚.
• If m∠ABC=180˚ and x=12, what is m∠GBH?
*ANSWER: ∠GBH is 30˚.
QuickTime™ and a
decompressor
are needed to see this picture.
• If m∠KML=55˚ and m∠JML=140˚, what is m∠JMK??
*ANSWER: ∠JMK is 85˚.
QuickTi me™ and a
decompressor
are needed to see thi s pi ctur e.
Links to Help You Study
• These links are all about the angle addition postulate.
• http://www.icoachmath.com/SiteMap/AngleAdditionP
ostulate.html
• http://www.rio.k12.wi.us/MATH/geo1.html
• http://www.schooltube.com/video/3322d1fcecd1487d
8e40/Angle-Addition-Postulate
By Jess Van Saders
Biconditional Statements
Stephanie Friedkin
What is a Biconditional Statement?
•
The statement is usually represented by
•
If a conditional statement & its converse
are both true, then it can be written as a
biconditional statement, using “if and only
if”.
pq
Examples
1.
Conditional: If the sum of two angle measures is 180 degrees, then the angles are
supplementary. √
Converse: If the angles are supplementary, then the sum of two angle measures is 180
degrees. √
The sum of two angle measures is 180 degrees if and only if the angles are
supplementary.
2.
Conditional: If a triangle is isosceles, then the triangle has two congruent sides. √
Converse: If a triangle has 2 congruent sides, then the triangle is isosceles. √
A triangle is isosceles if and only if the triangle has two congruent sides.
3.
Conditional: If a candidate becomes president, they have won the election. √
Converse: If a candidate wins the election, they become president. √
A candidate becomes president if and only if they win the election.
Helpful Links
• Tutor Vista: http://www.tutorvista.com/content/math/booleanalgebra/boolean-algebra/conditional-biconditionalstatements.php
• Regents Prep:
http://www.regentsprep.org/Regents/math/geometry/GP1/bic
on.htm
• Math Cuer:
http://mcuer.blogspot.com/2007/07/geometry-chapter22-definitions-and.html
Inductive Reasoning
by Amanda Mezzina
•
•
The basics of geometry began when people started to recognize patterns. In
fact, looking for patterns and making conjectures is part of the inductive
reasoning process.
The goal of inductive reasoning is to be able to find and describe patterns, and
to make real-life conjectures using this reasoning.
Using Inductive
Reasoning
Finding and Describing Patterns
Goal 1
1. Sketch the next figure in the pattern.
•
There are three simple
steps
Goal
2 of inductive
reasoning.
– 1. Look for a pattern. Look at several
examples. Use diagrams and table to help
discover a pattern.
Each figure in the pattern looks like the previous – 2. Make a Conjecture. Use example to make a
conjecture. A conjecture is an unproven
figure with another row of squares added to the
statement based on observations. Discuss the
bottom. Each figure looks like a stairway.
conjecture with other, and if necessary modify
it.
– 3. Verify the Conjecture. Use logical reasoning
to verify that the conjecture is true in all cases.
Extra Examples
1.
32
42
52
Each row is adding another square.
2. 1,4,16,64
-->
3. -5, -2, 4, 13
-->
Each number is four times the
previous number. The next number
is 256.
You add 3 to get the second
number, then 6 to the third number,
and you continue to add the next
multiple of three.
Need more help?
• http://www.basic-mathematics.com/examples-ofinductive-reasoning.html
• http://www.tutorvista.com/math/inductivereasoning-in-geometry
• http://www.tutorvista.com/math/inductivereasoning-in-geometry
Finding the midpoint of a
segment on a coordinate
plane
Casey Spillane
Period 2
1/20/11
The formula for finding the midpoint of a
segment on a coordinate plane is relatively
easy.
It involves taking two points on a coordinate
plane and finding the distance between them.
Formula
• You take the two endpoint of the
segment (x1, y1) (x2, y2)
• And add the Xs together and divide the
sum by 2, then do the same for the Ys.
That should give you the coordinates of
a point directly in the middle of the two
Practice Problems
•
Find the distance between the two points:
1) (3,-4) (6,8)
Answer: (4.5, 2)
2)
(10,-7) (5,4)
Answer: (7.5, 1.5)
3)
(3,9) (14,7)
Answer: (8.5, 8)
Ex:
Web Links
• http://www.mathopenref.com/coordmidp
oint.html
• http://www.onlinemathlearning.com/coor
dinate-geometry.html
• http://hotmath.com/help/gt/genericalg1/s
ection_12_3.html
The Law of Detachment
By Kyle Sorreta
The Law of Detachment
The law of detachment is used for
determining if a conditional statement is
true or not. In this power-point I will be
explaining the law, showing some
examples, and giving some helpful
websites.
Rules, Properties, and
Formulas
Rule: The law states that if P=Q then the
value of P is also the value of Q
Examples
• If two angles equal 90º, then they are
complementary. Angle 1 and angle 2 add up
to 90º. So angle 1 and angle 2 are
complementary.
• Kelly knows that if she misses practice before
a game, then she will not start in the game.
Kelly misses practice, so she concludes that
she will not start.
• Jim only eats sandwiches on Tuesdays. It is
Tuesday , so Jim will eat a sandwich.
Helpful Sites
• http://tutorusa.com/free/geometry/worksheet/deductivereasoning-law-detachment-law-syllogism
• http://snippets.com/what-is-the-law-of-detachment-ingeometry.htm
• http://scienceray.com/mathematics/geometry/geometr
y-help-deductive-reasoning/
Slope Intercept Form
•It is probably one of the most frequently
used ways to express the equation of a
line.
•One goal is to be able to find the slope
of a line when you are given two
coordinates.
•We also are trying to find the yintercept.
Important Rules
• Formula for y intercept: y=mx+b
• M is the slope of the line and b is the y- intercept.
• When given two points of a line, the first thing you should find is
the slope. You would find it using the slope formula. y2 - y1
x2 - x1
• Then once you’ve found m, all you have to do is plug it into the y
intercept equation using an x and a y from one of the
coordinates that they give you.
Example Problems
1. (-2, 8) and (6, 12)
12 -8
6-2
4
12= 1(6)+b
b=6
Y= 1x+6
2. (0, 16) and (-8, 22)
22-16
6
-8 - 0 -8
16= 3/-4(0)+b
b=16
Y= 3/-4x+16
3. (2, 4) and (1, -2)
-2-4
m=6
1-2
4= 6(2)+b
b= -8
Y= -8x+b
4
m=1
m= 3/-4
Helpful Sites
• http://www.purplemath.com/modules/slopgrph.htm
• http://www.purplemath.com/modules/strtlneq.htm
• http://www.glencoe.com/sec/math/algebra/algebra1/al
gebra1_05/study_guide/pdfs/alg1_pssg_G041.pdf
Summary
• When you graph lines on a coordinate
plane you’re given equations to use.
• The goal of this presentation is to teach
you or remind you of how to graph the
lines with these equations.
•
•
•
•
Equations
of
lines
An equation of a line is usually in slope intercept form (y=mx+b)
y= point on y-axis, x= point on x-axis, m= slope, b= y-intercept
So an equation would look like y=2x+1
To graph this line you would start with the y-intercept (+1) and put a point on the
y-axis. Then you would follow the slope (2) which equals to 2/1. So you would
move to spaces up and then one space to the right and make a point there.
Continue to make points one after another. You can also go backwards if there
isn’t enough room, 2/1 = down two and to the left one. Once you have your points
you draw your line.
In the picture, A is the y-intercept and the
other points follow the slope.
Examples
•
•
y=2/3x-3
Answer:
•
•
y=5
Answer:
•
•
2x-3y=6
Answer:
Extra Help
• http://www.purplemath.com/modules/slo
pgrph.htm
• http://www.math.com/school/subject2/le
ssons/S2U4L2GL.html
• http://www.algebralab.org/lessons/lesso
n.aspx?file=algebra_lineareqintercepts.
xml
Vertical Angle Theorem
By Chrissy Loganchuk
Summary
The goal of this is to explain what the
vertical angle theorem states and
examples of when you would use the
vertical angle theorem.
What is the Vertical Angle
Theorem?
If two angles are vertical angles their sides
form two pairs of opposite rays.
The theorem states that vertical angles are
congruent.
Problems
What is the measure of
=77.61
What is the measure of
=102.39
What is the measure of
=47.27
?
?
?
Links
http://library.thinkquest.org/2647/geometry
/angle/proof2.htm
http://hotmath.com/hotmath_help/topics/v
ertical-angles-theorem.html
http://www.mathwarehouse.com/geometry
/angle/vertical-angles.php
CONVERSE ALTERNATE
INTERIOR ANGLE THEOREM
Marissa Silverman
Summary & Goals
• The alternate interior angle theorem
states “if alternate interior angles are
congruent then the two lines are
parallel.”
• The goal of this is to find out if the two
lines are parallel
Rules & Properties
(More Like Hints)
• If you see a “z” and the angles inside of
it are congruent the lines are parallel
• That means that you are using the
converse alternate interior angle
theorem
Examples
Are these lines
parallel?
Yes they are. There are arc on
the angles show that the angles
are congruent.
Are these lines parallel?
Yes they are. They angles are
congruent.
Are these lines parallel?
Yes because the alternate interior
angles are congruent.
Web Links
• http://library.thinkquest.org/20991/geo/p
arallel.html
Congruent Triangle Proof
QuickTime™ and a
decompressor
are needed to see this picture.
By: Charlie Marshall
Summary
Goal: To prove 2 triangles are congruent,
using a proof table and multiple thereoms,
postulates, and laws.
The main idea of congruent triangle proofs is
to use a proof table and show that two
triangles are congruent using congruence
shortcuts such as: ASA, AAS, SAS, SSS,
HL
•
•
Rules Properties and
Formulas
There are 5 postulates used in finding congruence. The one shown is ASA, or
Angle Side Angle. When a triangle fits the descriptions of one of the congruence
postulates, you can mathematically state it is congruent.
List of Postulates:
– ASA- states that if two angles and the included side of one triangle are
congruent to two angles and the included side of another triangle, then these
two triangles are congruent.
– SAS- states that if two sides and the included angle of one triangle are
ccongruent to two sides and the included angle of another triangle, then these
two trianges are congruent.
– SSS-states that if three sides of one triangle are congruent to three sides of
another triangle, then these two triangles are congruent.
– AAS-states that if two angles and the non-included side one triangle are
congruent to two angles and the non-included angle of another triangle, then
these two triangles are congruent.
– HL- states If the hypotenuse and leg of one right triangle are
congruent to the corresponding parts of another right triangle, then
the triangles are congruent
QuickTime™ and a
decompressor
are needed to see this picture.
Examples
AB and XY are congruent hypotenuses
QuickTime™ and a
decompressor
are needed to see this picture.
1.
2.
3.
4.
AB = XY
<C = 90°,
<Y = 90°
AC = ZY
∆ABC = ∆ZYX
For HL, you must come up with a ASS or SSA
postulate. These technically only work on right
triangles. So the reason the last statement in
this proof is HL, is that it’s a right triangle
1.
2.
3.
4.
Given
Definition of a right angle
Converse Pythagorean theorem
HL
Remember that bisects splits and angle in half.
QuickTime™ and a
decompressor
are needed to see this picture.
1.
2.
3.
<OMN = <OML;
OM = OM
∆OMN = ∆OML
<LOM = <NOM
1.
2.
3.
Given
Reflexive Property
ASA
Even though the sides are touching, it doesn't
mean the side is congruent.
QuickTime™ and a
JK = MN; Triangles are
isosceles
decompressor
are needed to see this picture.
1.
JK = MN,
2.
<KJL = <NMO, <KLJ = <NOM
3.
∆KLJ = ∆NOM
1.
2.
3.
Given
Definition of an isosceles triangle.
AAS
WEBSITES
• http://www.mathwarehouse.com/geomet
ry/congruent_triangles/
• http://www.mathwarehouse.com/geomet
ry/congruent_triangles/isoscelestriangle-theorems-proofs.php