geometry-definitions, postulates, properties, & theorems ch 3-perpendicular & parallel lines...

Geometry-Definitions, Postulates, Properties, &

TheoremsCH 3-Perpendicular & Parallel Lines

Geometry I

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PostulatesTheorems Theorems Cont.

Basic Definition

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Quiz (end of class)

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Quizlet(HW)

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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

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each of the sections of the PowerPoint (they are abbreviated).

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zlet

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T

TC

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H

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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

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*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 ?

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*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