Warm-up Open book to page 73 Read all of the 3.1 section

Warm-up Open book to page 73 Read all of the 3.1 section
Some questions to think about… What is the difference between parallel and skew? Can segments and rays be parallel? What can 2 planes be? Parallel? And? What can a line and a plane be?

Chapter 3 Parallel Lines and Planes
Learn about parallel line relationships Prove lines parallel Describe angle relationship in polygons

3.1 Definitions Objectives State the definition of parallel lines
Describe a transversal

Parallel Lines ( || ) Coplanar lines that do not intersect. m || n m n
The only difference between skew and parallel is the coplanar part. What are the other ways to define parallel? n

Parallel Lines ( or ) The way that we mark that two lines are parallel is by putting arrows on the lines. m || n m The only difference between skew and parallel is the coplanar part. What are the other ways to define parallel? n

Skew Lines ( no symbol  )
Non-coplanar lines, that do not intersect. p q The only difference between skew and parallel is the coplanar part. What are the other ways to define parallel? What is the difference between the definition of parallel and skew lines?

Can a plane and a line be parallel?
Parallel Planes Planes that do not intersect. P Q A plane and a line can be parallel. Any line contained in plane P would be parallel to plane Q. Can a plane and a line be parallel?

Parallel, intersecting , or skew?
Name parallel lines Name skew lines Names 5 lines parallel to plane ABCD Name parallel planes

Theorem B P A D Q C 2 levels of a parking structure are parallel
They need to be reinforced by support beams (AB and CD) What do you think you can say about the beams based on what you know about the 2 levels (planes)? B P This is the first of six ways to prove lines are parallel. A D If two parallel planes are cut by a third plane, the the lines of intersection are parallel. Q C

The Transversal t r Any line that intersects two or more coplanar lines in different points. s

Name the transversal If j, k and l are coplanar name the transversal.

Understanding the position of these special angles is key!!
Special Angle Pairs exterior Understanding the position of these special angles is key!! t 1 2 3 interior 4 r 5 6 s These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. When dealing with parallel lines, the relationships become more specific 7 8 exterior

Corresponding Angles (corr. s)
t Think - shape 1 2 3 4 r 5 6 s These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. 7 8

Corresponding Angles 1 and  5 3 and  7 2 and  6 4 and  8 t 1 2
These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. 7 8

Alternate Interior Angles (alt. int. s)
Think - shape 1 2 3 4 r 5 6 s These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. 7 8

Alternate Interior Angles
 4 and  5 3 and  6 1 2 3 4 r 5 6 s These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. 7 8

Same Side Interior Angles (s-s. int s)
Think - shape 1 2 3 4 r 5 6 s These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. 7 8

Same Side Interior Angles
 4 and  6  3 and  5 1 2 3 4 r 5 6 s These are just labels for angle relationships. When the lines are parallel, the relationships become more specific. Although not covered by the textbook, an angle pair such as 1 and 8 could be called alternate exterior without causing any confusion. 7 8

**TRANSVERSAL** t What is unique about the transversal in relation to any of the special angle pairs? The transversal line will always be a part of making up BOTH ANGLES. 1 2 3 4 r 5 6 s 7 8

Group Practice Name the two lines and the transversal that form each pair of angles. Re-draw the diagram with only the lines you need Label the “exterior” and “Interior” areas of the diagram k l n t 3 4 1 2 5 6 7 8 9 10 11 12 15 16 13 14 **THE TRANSVERSAL IS ALWAYS THE LINE THAT HELPS CREATE THE PAIR OF ANGLES**

 1 and  3 k l n t 3 4 1 2 5 6 7 8 9 10 11 12 15 16 13 14

 3 and  11 k l n t 3 4 1 2 5 6 7 8 9 10 11 12 15 16 13 14

 10 and  11 k l n t 3 4 1 2 5 6 7 8 9 10 11 12 15 16 13 14

 6 and  9 k l n t 3 4 1 2 5 6 7 8 9 10 11 12 15 16 13 14

 8 and  11 k l n t 3 4 1 2 5 6 7 8 9 10 11 12 15 16 13 14

Remote Time

True or False A transversal intersects only parallel lines.

True or False Skew lines are not coplanar

True or False If two lines are coplanar, then they are parallel.

True or False If two lines are parallel, then exactly one plane contains them.

(A) Alternate interior angles (B) Same-side interior angles (C) Corresponding angles
 2 and  7 1 4 3 2 5 6 7 8 I B F C H G A D E

 ADE and  DEB 1 4 3 2 5 6 7 8 I B F C H G A D E

 BEF and  EGI 1 4 3 2 5 6 7 8 I B F C H G A D E

3.2 Properties of Parallel Lines
Objectives Learn the special angle relationships …when lines are parallel Pictured: The Georges Pompidou Center. Paris, France

“IF” - what we know Lesson Focus ‘Then” – what we can prove
In this lesson we know that we have 2 parallel lines being cut by a transversal

If 2 || lines are CBT, then….
Corresponding angles are congruent corr. Ls are  Alternate interior angles are congruent alt. int. Ls are  Same-side interior angles are supplementary S-S int. Ls are Supp.

IF YOU ONLY REMEMBER 1 THING….
When 2 parallel lines are CBT the whole diagram will only have 2 different angle measurements!!!

Name all angles that are congruent to  1
2 3 4 5 6 7 8

Name all angles that are supplementary to  1
2 3 4 5 6 7 8

If m  5 = 60, then m  4 =_____ and m  2 = _____
1 2 3 4 5 6 7 8

If m  7 = 110, then m  3 =_____ and m  4 = _____
2 3 4 5 6 7 8

Group work Complete the proofs for alt. interior angles

Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. r s t 1 2 3 4 5 6 7 8 If 2 || lines are CBT, the corr. Ls are congruent Postulate 10 cannot be proven, because it is the first angle relationship and so is assumed to be true. The other angle relationships can be proven, because of the assumption that corresponding angles are congruent. Either of the other two angle pairs could have been selected to serve as the assumption, and the others proven from that…the key here is that the choice was arbitrary. Can you name the corresponding angles?

Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. r s t 1 2 3 4 5 6 7 8 If 2 || lines are CBT, then alt. int. Ls are congruent Prove theorem 3-2 with them. Can you name the alt. int. angles?

Theorem If two parallel lines are cut by a transversal, then same side interior angles are supplementary. r s t 1 2 3 4 5 6 7 8 If 2 || lines are CBT, then s-s int. Ls are supp. Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way. Can you name the s-s int angles?

Theorem A line perpendicular to one of two parallel lines is perpendicular to the other. t r *Based on the position of these angles what did we already learn in this lesson that told us this was true? No need to prove this one. s

Group Practice Find the values of x and y 80º 2yº xº

Group Practice Find the values of x and y xº 3yº

Group Practice Find the values of x and y xº yº 50º 60º

Group Practice Find the values of x and y 110º xº 2yº

WARM-UP S-S Int Angle Proof

3.3 Proving Lines Parallel
Objectives Learn about ways to prove lines are parallel Use Theorems about parallel lines Define an auxiliary line This lecture presents statements that are the converses of the statements learned yesterday. Take some time to review converses and the importance of keeping the language, and the logic, straight.

How many red lines can you show as perpendicular?
How many of the red lines can you place going through the point that would be parallel the black line? How many red lines can you show as perpendicular? 55

In your notes Black and white postulate sheet Answer the following
What is the difference between post. 10 and 11? What is my evidence in each? What do I prove in each one? 3.2 – The parallel lines told us about the special angle relationships. 3.3 – Special angle relationship help us determine that the lines are parallel.

TELL ME WHY THE LINES ARE PARALLEL?
“The lines are parallel because the ________________________________________.” FILL IN THE BLANK WITH ANY OF THE FOLLOWING…. Corresponding angles are congruent Alternate interior angles are congruent Same side interior angles are supplementary 2 lines are perpendicular to the same line 2 lines are parallel to the same line

Postulate If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. If 2 lines are CBT and the corr. Ls are congruent, then the lines are ||. This postulate is the converse of post. 10 Since the direction is reversed, this postulate can be used to prove lines are parallel. All the statements today can be used to prove lines are parallel. 1 m 2 If 1  2, then m || n. n

Theorem If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. If 2 lines are CBT and the alt. int. Ls are congruent, then the lines are ||. Prove this with them. Note that none of the stuff learned yesterday can be used because they all start “in two parallel lines are cut by a transversal…” and in these proofs, the lines are not parallel, YET. Make sure to emphasize this! m 1 2 If 1  2, then m || n. n

Theorem If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel. If 2 lines are CBT and the s-s int. Ls are supp, then the lines are ||. Ditto. 1 m 2 If 1 suppl  2, then m || n. n

Theorem In a plane two lines perpendicular to the same line are parallel. t If t  m and t  n , then m || n. No proof needed on this one. m n

**THINK TRANSITIVE PROPERTY
Theorem Two lines parallel to the same line are parallel to each other p If p  m and m  n, then p  n This is the final way to prove lines are parallel. The next slide summarizes the ways. m **THINK TRANSITIVE PROPERTY n

Partner Work (skip) Complete proof of theorem 3-5

Experiment - Goal Find parallel lines

Materials 5 Pencils Paper

Which pair of lines are parallel?
“Line ___ and line ____ are parallel because the _______________________________.”

A  l & t B  j & k C  none Which pair of lines are parallel?
“Line ___ and line ____ are parallel because the _______________________________.” l t A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

m  6 = m  4 l t A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

m  2 + m  3= m  5 A  l & t B  j & k C  none l t 8 6 j 2 4 5 1 3
7 k

m  2 + m  3 + m  8 = 180 A  l & t B  j & k C  none l t 8 6 j 2 4
5 1 3 7 k

 7   1 l t A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

m  1 =m  8 = 75 l t A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

 5 and  6 are supplementary
A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

 4 and  5 are supplementary
A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

 2 and  3 are complementary and m  5 = 90
A  l & t B  j & k C  none 8 6 j 2 4 5 1 3 7 k

TELL ME WHY THE LINES ARE PARALLEL?
“The lines are parallel because the ________________________________________.” FILL IN THE BLANK WITH ANY OF THE FOLLOWING…. Corresponding angles are congruent Alternate interior angles are congruent Same side interior angles are supplementary 2 lines are perpendicular to the same line 2 lines are parallel to the same line 76

Ch. 3 QUIZ Definition of parallel and skew lines
When two parallel lines are cut by a transversal Name different types of pairs of angles Be able to name all the angles congruent to a certain angle Find measurements of angles (Go off the information given, not assumptions of the diagram.) Question directly from 3.3 worksheet Know the 5 ways to prove lines parallel

3.4 Angles of a Triangle Objectives Classify Triangles
State the Triangle Sum Theorem Apply the Exterior Angle Theorem

Draw a triangle in your notes
A triangle is a figure formed by three segments Joining noncollinear points. Not in a straight line Draw a triangle in your notes

Vertex Each of the three points is a vertex of the triangle. (plural vertices) A ▲ABC B C

Sides The segments are the sides of the triangle A AB AC BC C B

Types of Triangles (by sides)
WHAT DO YOU THINK ARE WAYS WE CAN CLASSIFY (NAME) TRIANGLES IN TERMS OF THEIR SIDES? Isosceles 2 congruent sides Equilateral All sides congruent Scalene No congruent sides Ask which pairs of these can be concurrent and which cannot.

Types of Triangles (by angles)
Can you name these triangles? Equiangular Ask which pairs of these can be concurrent and which cannot. Acute 3 acute angles Right 1 right angle Obtuse 1 obtuse angle

Experiment 1 GOAL: Find the sum of the angles of a triangle
Write down the goal of the experiment, answer the questions in your notes as we go along

Materials Paper Triangle Pencil

Procedure Label the angles of the triangle  1,  2 and  3
Draw a point at the tip of each vertex

3. Rip off the three angles of the triangle
1 3 2

4. Put the three angles in a row so that the angles meet at one point and at least one side of each angle touches the side of another angle 5. What is the sum of those angles?  Hint use the Angle Addition Postulate. 1

Something else to think about
Can you draw a triangle where the sum of the angles is not 180 degrees?

Theorem The sum of the measures of the angles of a triangle is 180
B mA + mB + mC = 180 Do the proof. This is the first proof that requires the drawing of an auxiliary line. When finished, show the alternative placement of the line. C A **This is true for any and all triangles!!!

Remote Time

A – Acute B – Obtuse C - Right
State whether a triangle with two angles having the given measures is acute, obtuse, or right.

55 43

47 43

Corollaries (pg.94) Corollaries are just like theorems, but are so closely related to a single theorem, that they are listed as being associated with that theorem. Just like theorems, they can be used as reasons in proofs THESE COROLLARIES ARE BASED OFF A TRIANGLE = 180.

Corollary 1. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are _________. Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Why? Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary 2. Each angle of an equiangular triangle has measure___.
Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary 2. Each angle of an equiangular triangle has measure 60º.
Why? Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary 3. In a triangle, there can be at most one _____ or _______ angle. Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary 3. In a triangle, there can be at most one right or obtuse angle. Why? Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary 4. The acute angles of a right triangle are _____________.
Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Corollary 4. The acute angles of a right triangle are complimentary.
Why? Corollaries are just like theorems, but are so closely related to a single theorem,that they are listed as being associated with that theorem. Select and prove one of these corollaries.

Find the measure of  A 80º 50º A 50º

Experiment 2 GOAL: Explore the exterior angles of a triangle

Materials Protractor Pencil Paper Ruler

Copy the diagram 1 2 3 4

One triangle has 6 different exterior angles
1 2 3 4  1 is an exterior angle One triangle has 6 different exterior angles

m1 + m 2 = 180 Why ? 3 Do the proof of this theorem. 1 4 2

they are ffffaaaaarrrr away from angle 1
 3 and  4 are called remote interior angles 3 2 1 4 they are ffffaaaaarrrr away from angle 1

it is right next to angle 1
2 3 4  2 is the adjacent interior angle

Do you notice any measurements that are equal? Any that add up to 180?
Measure the four angles of your triangle using the protractor. Be precise ! m  1 = m  2 = m  3 = m  4 = Do you notice any measurements that are equal? Any that add up to 180?

Theorem The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles m m = 180 m2 + m3 + m = 180 m1 = m3 + m4 3 Do the proof of this theorem. 1 4 2

Use what you know about parallel lines, triangles, and ext angles to solve….

3.5 Angles of a Polygon Objectives Define a regular and convex polygon
Calculate the interior and exterior angles of a polygon

Polygon Means “many angles”
What is a polygon? Why does the definition need to be so specific. Show examples that make this definition so necessary.

The Polygon 1. Each segment intersects exactly two other segments, one at each endpoint. 2. No two segments with a common endpoint are collinear What is a polygon? Why does the definition need to be so specific. Show examples that make this definition so necessary. **Your shape must enclose an area with its sides, therefore each side is connected, with no overlaps**

Not Polygons Can you explain why each of these figures is NOT a polygon ? What is a polygon? Why does the definition need to be so specific. Show examples that make this definition so necessary.

Not Polygons Not a segment

Not Polygons Doesn’t intersect 2 other segment one at each endpoint

Not Polygons Exactly 2 other segment

Convex Polygon No line containing a side of the polygon contains a point in the interior of the polygon. Basically, a convex polygon is one where if you extend the sides, the lines will never cross into the interior of the figure. Most of the theorems and properties covered about polygons only apply to convex polygons, not “dented” ones. Non -convex convex

The Diagonal A segment that joins non-consecutive vertices.
What polygon does not have any diagonals? Choose one “start” point What polygon does not have any diagonals? Since diagonals can only connect vertices that are not already connected, the answer if the triangle.

# of sides Picture Name

Names of Polygons Number of sides Name 3 Triangle 4 Quadrilateral
Pentagon Hexagon Octagon Decagon n n-gon Talk about the correspondence between the names of the polygons and the names of the months of the year. September, October, November and December used to be the seventh, eighth, ninth and tenth months of the year. Origin of the names of the months of the year can be found at

How many triangles do you see ?
# of sides Picture Name # of triangles Choose one point on the polygon and draw all the possible diagonals from that point. How many triangles do you see ?

One triangle has an angle sum of 180.
# of sides Picture Name # of triangles Interior angle sum One triangle has an angle sum of 180. What is the angle sum inside the polygon?

(n-2)180 How did we come up with this formula? Look at your tables
What do you notice about the number of sides compared to the number of triangles that can be drawn from the diagonals? 4 sides 2 triangles =(4-2)180 =(2) 180 5 sides 3 triangles =(5-2)180 =(3) 180

Regular Polygon (p.103) All angles congruent All sides congruent
Although both of these formulas are useful, the first one is arithmetically complicated, especially without a calculator. Because each exterior angle is supplementary to an interior angle, work through the exterior angles to find the interior angles…its formula is much easier.

REGULAR POLYGON # of sides Picture Name # of triangles
Interior angle sum One interior angle One exterior angle Exterior angle sum

Exterior Angle + Interior Angle =
180 Backside of worksheet

Theorem The sum of the measures of the exterior angles, one at each vertex, of a convex polygon is 360. 4 1 1 3 Ask them to take an imaginary walk around the perimeter of of these shapes, as if they are laid out on a large field (think alien figures as wheat field crop patterns). If they start anywhere on the figure and walk its perimeter, they will have turned through each exterior angle as they approach each corner in order to complete the circuit. Since they walked in a full circle, they turned through 360 degrees. The result is the same no matter how many sides the polygon has! 3 2 2 1 + 2 + 3 = 360 1 + 2 + 3 + 4 = 360 **THIS APPLIES TO ANY CONVEX POLYGON!!

**The only time you can divide by n is when you have a regular poly.
REGULAR POLYGONS All the interior angles are congruent All of the exterior angles are congruent 360 = the measure of each exterior angle n **The only time you can divide by n is when you have a regular poly.

Remote Time A – Always B – Sometimes C – Never

A – Always B – Sometimes C – Never
The sum of the measures of the exterior angles of any polygon one angle at each vertex is ___________ 360º

The sum of the measures of the interior angles of a convex polygon is ____________ 360º

The sum of the measures of the exterior angles of a polygon __________ depends of the number of sides of a polygon.

White Board Find the interior angle sum
Heptagon Find the measure of one interior and exterior angle Regular Decagon

White Board An exterior angle of a regular polygon has measure 10. The polygon has _____ sides. 36 sides 142

White Board An interior angle of a regular polygon has measure The polygon has _____ sides. 18 sides 143

White Board Three of the angles of a quadrilateral have measures 90, 60, and The fourth angle has measure ___ . 95 144

3.6 Inductive Reasoning Objectives
Discover the uses and hidden dangers of inductive logic Compare inductive and deductive logic

Inductive Logic The conclusion is based on seeing a pattern from observations. Is probably true, but need not be. Can be dangerous because of its tendency to allow generalization from specific information. Inductive logic is how many great discoveries are made. Archimedes is pictured. Archimedes

Deductive Logic Based on accepted statements Conclusion must be true.
Definitions Postulates Theorems Etc… Conclusion must be true. Two-column proofs are deductive. Deductive logic is how things are definitively proven. Pythagoras is pictured. Pythagoras

Remote time A – Inductive Reasoning B – Deductive Reasoning

A – Inductive Reasoning B – Deductive Reasoning
Mary has given Jimmy a present on each of his birthdays. He reasons that she will give him a present on his next birthday.

John knows that multiplying a number by -1 changes the sign of the number. He reasons that multiplying a number by an even power of -1 will change the sign of the number an even number of times. He concludes that this is equivalent to multiplying a number by +1.

Ramon noticed that sloppy joes had been on the school menu the past 5 Fridays. Ramon decides that the school always serves sloppy joes on Fridays.

White Board Look for a pattern and predict the next two numbers in each sequence.

1, 1, 2, 3, 5, ___, ___

1, 1, 2, 3, 5, 8, 13

1,3,5,___,___,

1, 3, 5, 7, 9

12, 7, 2, ___, ___

12, 7, 2, -3, -8

Ch. 3 test Study and understand all of the postulates in the chapter – you will have to fill in the blanks Definition of parallel and skew lines Understanding the formula used for the interior and exterior angles of a polygon How can you use it to find the measurement of one interior angle of a regular polygon Use it to find how many sides a polygon is based off the exterior angle measurement

Given a diagram – solve for the given angles and variables
**NEED TO BE ABLE TO DISTINGUISH WHICH TWO LINES ARE PARALLEL AND WHICH LINE IS TRANSVERSAL – HOW IS EACH ANGLE BEING FORMED? Other things to help solve angles and variables.. Is a bigger angle made up of two smaller ones?? Vertical angles Supplementary angles The sum of a triangle is 180 Exterior angle = 2 remote interior angles

Using inductive reasoning to solve a number pattern
Being able to prove which lines parallel based off of certain angles being congruent How many lines can be parallel and perpendicular to a point outside a line Write the postulate or theorem word for word Hint: it will be the ones form 3.2 or 3.3 Fill in and writing an entire proof

Warm-up Open book to page 73 Read all of the 3.1 section

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