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Geometry-Definitions, Postulates, Properties, &
TheoremsCH 3-Perpendicular & Parallel Lines
Geometry I
Takes you to the main menu
Takes you to the help page
There are other buttons explained throughout the PowerPoint
Navigating the PowerPoint
Main Menu
Background Info
PostulatesTheorems Theorems Cont.
Basic Definition
s
Quiz (end of class)
Quiz Info(next day)
Quizlet(HW)
Helppp!!
Click the buttons to navigate to the different slides(It’s best to follow the arrows)
✪
A theorem is a statement that has been proven on previously established statements like other theorems and postulates.
A theorem is a proof of the truth of the resulting expression.
A theorem is a logical argument in the sense that if a hypothesis is true then the conclusion must also be true.
A postulate is a statement that is accepted without proof and is fundamental to a subject
Background Info
NOTE: Click each theorem to take you to another slide with examples
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary
If two lines are perpendicular, then they intersect to form four right angles
Theorems
Alternate Interior Angles: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Consecutive Interior Angles: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary
Alternate Exterior Angles: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent
Theorems Cont.
Note: Click each postulate to go to another slide with examples
Parallel Postulate: 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
Perpendicular Postulate: 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
Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Postulates
NOTE: Click each definition to be directed to another slide with examples
Parallel Lines-Two lines that are coplanar and do not intersect
Skew Lines- Two lines that do not intersect and are not coplanar
Transversal- A line that intersects two or more coplanar lines at different points
Perpendicular Lines- Two lines intersect to form a right angle
Basic Definitions
Homework (Print off a copy to bring to class) Go to quizlet.com Select create in the upper left-hand corner Make flashcards Create an account View the extra features Study your flashcards for the quiz tomorrow
Quizlet
Out of the 13 definitions only 5 will be on the quiz tomorrow. A mixture of the names and definitions will be placed in a cup. From there, you and your classmates will come to the front and choose one from the cup. If the name or part of the definition is drawn, the definition must be written (Pretty close to word-for-word). If the definition is drawn, the name must be written. After the 5, students can raise their hands if they wish to continue with the quiz. It only takes 1 to force the class to answer more.
Quiz Info
Helppp!!
Ask peers Read through book Look on YouTube or Khan Academy Come see me before or after school
Click the buttons around the slide for a description of
each of the sections of the PowerPoint (they are abbreviated).
R
Qui-
zlet
QI
Q
BI
P
T
TC
BD
H
✪
This slide explains the definitions of theorems and postulates.
BI (Background Info)
The first three of the chapter are given along with their definitions.
Click the postulate to take you to another slide where examples are presented.
P (Postulates)
The first six theorems of the chapter are given along with their definitions.
Click the theorem to take you to another slide where examples are presented.
T (Theorems)
This slide is a continuation of the previous theorems slide.
TC (Theorems Cont.)
Four definitions are given. Buttons take you to another slide where visuals are displayed.
BD (Basic Definitions)
This slides provides a link to take you to the end of the presentation where there will be a short quiz.
The quiz will not be graded, but try your best because it is great practice for the quiz tomorrow!
Q (Quiz)
This slide provides information on and the expectations for the quiz the following day.
QI (Quiz Info)
This slide provides a brief explanation on how to use Quizlet and what is expected for the homework for the following day.
Qui-zlet (Quizlet)
This slide presents ways to get extra assistance if needed.
It also provides buttons for each of the sections, which take you to another slide that describes each of the sections.
The titles of the buttons are abbreviated by the first letter of each word.
H (Helppp!!)
This theorem is quite significant in geometry. We will use it during proofs. Click the picture below to see an example of a proof that we will use this theorem for.
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular
Click Theoremto returnto Theorems
Below is a link to a video of a proof that we will solve in this class
Video Proof
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary
Click TheoremTo return to Theorems
Using this theorem and the definitions of perpendicular lines and right angles, you will be able to write different kinds of proofs for the same situation.
If two lines are perpendicular, then they intersect to form four right angles
Proof Hints:• Use definition of perpendicular
lines to find one right angle.• Use vertical and linear pairs of
angles to find three more right angles.
Click Theorem to return to Theorems
Click on the link below to watch a video on the proof of the theorem.
AIA
Alternate Interior Angles: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are
congruent.
Click TC to return to Theorems Cont.
Click on the link to watch a video on a proof CIA
Consecutive Interior Angles: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are
supplementary
Click TC to return to Theorems Cont.
Alternate Exterior Angles: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are
congruent
Click TC to return to Theorems Cont.
Parallel Postulate: 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
Click Postulate to return to Postulates
Perpendicular Postulate: 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
If we wanted to measure the distance between a point and a line, we can employ the Perpendicular Postulate, a compass, and a straightedge, since there exists only one perpendicular line from a point to a line. Click the button below for an example:
Click Postulate toreturn to Postulates
So, in the figure below, if l || m, then angle 1 is congruent to angle 2.
Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are
congruent.
Click Postulate to return to Postulates
Parallel Lines-Two lines that are coplanar and do not intersect
Click this button!!!
Click BD to return to Basic Definitions.
Skew Lines- Two lines that do not intersect and are not coplanar
Click BD to return to Basic Definitions
Transversal- A line that intersects two or more coplanar lines at different points
Click BD to return to Basic Definitions
Perpendicular Lines- Two lines intersect to form a right angle
Click the button!!Click BD to return to Basic Definitions
Click this button to return to the theorem
The next few slides are a short quiz that will not be graded, but it is great practice for tomorrow’s quiz!
This quiz is meant to be completed after you have gone through the full PowerPoint.
You must answer the questions correctly before moving on to the next question
Click the Start button to begin
Start
True or False: This is the correct definition of parallel lines:Two lines are parallel if and only if they do not intersect. Click True or False TrueFalse
Question 1
Return to ? Move to next ?
Return to the ?
*Hint: There is a key word missing from the definition. Think about the definition of skew lines and how the two differ.
True or False:A postulate is a statement that is accepted without proof and is fundamental to a subject
TrueFalse
Question 2
Next ?Return to the ?
Return to the ?
*Hint: You’ll find the answer on the Background Info slide
Fill in the blank:If two sides of two adjacent acute angles are perpendicular, then the angles are ________
A. ComplementaryB. SupplementaryC. Obtuse
Question 3
Next ?Return to the ?
Return to the ?
*Hint: Refer to the Theorems slide
Which can you add to the blanks to make the statement correct?
If two parallel lines are cut by a transversal, then the pairs of _____ _____ angles are congruent.I. Consecutive InteriorII. Alternate ExteriorIII. Alternate Interior
A. I. and II.B. II. and III.C. All three
Question 4
Next ?Return to the ?
Return to the ?
*Hint: Check out the theorems
What makes the following statement incorrect?
A theorem is a statement that hasn’t been proven on previously established statements like other theorems and postulates.A. ProvenB. StatementsC. Hasn’t
Question 5
EndReturn to the ?
Return to the ?
*Hint: Is a theorem a proven statement?
You have completed the
lesson! Remember to study for your
quiz!
References
http://www.k6-geometric-shapes.com/geometric-terms.html http://etc.usf.edu/clipart/76000/76015/76015_3_d_lines.htm http://www.onlinemathlearning.com/pairs-of-lines.html http://www.themathlab.com/dictionary/swords/swords.htm http://www.mathsisfun.com/definitions/perpendicular-lines.html http://www.quia.com/jg/1076804list.html http://math.tutorvista.com/geometry/transversal-lines.html http://www.ducksters.com/kidsmath/angles_glossary.php http://www.mathsisfun.com/geometry/construct-perponline.html http://www.mathsisfun.com/geometry/construct-perpnotline.html http://www.mathsisfun.com/geometry/construct-paranotline.html http://www.docstoc.com/docs/120074625/Given-Angles-1_amp2-are-a-linear-pair-Angles-1_amp2-are-congruent
References Cont.
http://grotongeometry.wikispaces.com/Friday+February+29--Version+B http://mrgiomini.blogspot.com/2013/01/class-topics-010913.html http://www.sophia.org/proving-angles-are-complementary-with-perpendicular-lines/proving-angles-are-complementary-with-perpendicula--5-tutorial http://www.wyzant.com/help/math/geometry/lines_and_angles/angle_theorems http://www.youtube.com/watch?v=svprkO5bM88 http://www.mathsisfun.com/geometry/consecutive-interior-angles.html http://mrwadeturner.pbworks.com/w/page/26527656/Consecutive%20and%20Alternate%20Exterior%20Angles http://www.youtube.com/watch?v=GcyIDwMe808 http://doversherborn.comcastbiz.net/highschool/academics/math/baroody/GeometryHonors/Class%20Notes/Chapter%205/Lesson5-3/Lesson5-3.html http://hotmath.com/hotmath_help/topics/corresponding-angles-postulate.html http://en.wikipedia.org/wiki/Parallel_postulate
References Cont.
http://www.goodlightscraps.com/congrats.php http://www.noodle.org/learn/details/231602/example-of-something-that-goes-wrong-in-the-lab http://www.sodahead.com/living/can-you-please-tell-me-something-about-canada/question-1756107/?link=ibaf&q=wrong+answer http://www.broward.org/Kids/EnvironmentalKidsClub/Pages/WrongAnswer.aspx http://pyhong.blogspot.com/2010/03/thumbs-down-to-me.html http://tracemypreps.com/2013/01/03/review-of-goals-for-2012/stop-go-back-try-again/ http://www.keepcalm-o-matic.co.uk/p/keep-calm-and-try-again-36/
References Cont.
http://www.lavenir-jit.com/quiz-competition.html http://www.funscrape.com/Scraps.aspx?CatName=Good+Job http://www.grandprofile.com/Myspace_Comments/Congratulations_Comments/ http://www.clker.com/clipart-correct-mark-4.html https://smarttan.com/news/index.php/headline-story-smart-tan-member-responds-in-print-on-the-radio-to-false-iarc-report/ http://phsmath.pbworks.com/w/page/16606090/FrontPage http://support.prometheanplanet.com/server.php?show=nav.17545 http://www.gpsmagazine.com/2009/11/tomtom_one_140_s_review.php?page=4 http://simonhowells.typepad.com/my-blog/2010/02/new-blogpost-quizlet.html http://berkscountynbems.homestead.com/Berks-NBEMS-Help-Page.html