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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 Parallel Lines ( or ) The way that we mark that two lines are parallel is by putting arrows on the lines. m || n m n Skew Lines ( no symbol ) Non-coplanar lines, that do not intersect. q p What is the difference between the definition of parallel and skew lines? Parallel Planes Planes that do not intersect. P Q Can a plane and a line be parallel? 1. 2. 3. 4. 5. Parallel, intersecting , or skew? Name parallel lines Name skew lines Names 5 lines parallel to plane ABCD Name parallel planes Theorem 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 A If two parallel planes are cut by a third plane, the the lines of intersection are parallel. Q C D The Transversal t r s Any line that intersects two or more coplanar lines in different points. Name the transversal • If j, k and l are coplanar name the transversal. l j k Name the transversal • If j, k and l are coplanar name the transversal. k j l Name the transversal • If j, k and l are coplanar name the transversal. j k l Special Angle Pairs exterior t 1 2 3 interior 5 7 4 6 8 exterior r s Understanding the position of these special angles is key!! When dealing with parallel lines, the relationships become more specific Corresponding Angles (corr. s) t Think 1 2 3 4 r - shape 5 7 6 8 s Corresponding Angles t 1 2 3 5 7 4 6 8 r s 1 and 5 3 and 7 2 and 6 4 and 8 Alternate Interior Angles (alt. int. s) • Think t 1 2 3 4 r - shape 5 7 6 8 s Alternate Interior Angles t 4 and 5 3 and 6 1 2 3 5 7 4 6 8 r s Same Side Interior Angles (s-s. int s) • Think t 1 2 3 4 r - shape 5 7 6 8 s Same Side Interior Angles 4 and 6 3 and 5 t 1 2 3 5 7 4 6 8 r s **TRANSVERSAL** • What is unique about the transversal in relation to any of the special angle r pairs? t 1 2 3 5 7 4 6 8 s • The transversal line will always be a part of making up BOTH ANGLES. 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 1 5 k 2 6 9 10 13 14 l 4 3 7 11 12 15 16 t 8 n **THE TRANSVERSAL IS ALWAYS THE LINE THAT HELPS CREATE THE PAIR OF ANGLES** 1 and 3 k 1 2 5 6 9 10 13 14 l 4 3 7 11 12 15 16 t 8 n 3 and 11 k 1 2 5 6 9 10 13 14 l 4 3 7 11 12 15 16 8 t n 10 and 11 k 1 2 5 6 9 10 13 14 l 4 3 7 11 12 15 16 t 8 n 6 and 9 k 1 2 5 6 9 10 13 14 l 4 3 7 11 12 15 16 t 8 n 8 and 11 k 1 2 5 6 9 10 13 14 l 4 3 7 11 12 15 16 t 8 n 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 1 2 5 3 6 7 A 2 and 7 4 8 B D E C F G H I (A) Alternate interior angles (B) Same-side interior angles ADE and DEB (C) Corresponding angles 1 2 5 3 6 7 A 4 8 B D E C F G H I (A) Alternate interior angles (B) Same-side interior angles BEF and EGI (C) Corresponding angles 1 2 5 3 6 7 A 4 8 B D E C F G H I 3.2 Properties of Parallel Lines Objectives • Learn the special angle relationships …when lines are parallel Lesson Focus • “IF” - what we know • ‘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…. 1. Corresponding angles are congruent – corr. Ls are 2. Alternate interior angles are congruent – alt. int. Ls are 3. 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 1 2 3 4 5 6 7 8 Name all angles that are supplementary to 1 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 = _____ 1 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. t If 2 || lines are CBT, the corr. Ls are congruent r 1 2 3 4 Can you name the corresponding angles? 5 6 7 8 s Theorem If two parallel lines are cut by a transversal, then alternate interior angles t are congruent. r 1 2 If 2 || lines are CBT, then alt. int. Ls are congruent Can you name the alt. int. angles? 4 3 5 7 6 8 s Theorem If two parallel lines are cut by a transversal, then same side interior angles t are supplementary. r 1 2 If 2 || lines are CBT, then s-s int. Ls are supp. Can you name the s-s int angles? 4 3 5 7 6 8 s 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? 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 • 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? 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…. 1. Corresponding angles are congruent 2. Alternate interior angles are congruent 3. Same side interior angles are supplementary 4. 2 lines are perpendicular to the same line 5. 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 ||. 1 m 2 n If 1 2, then m || 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 ||. 2 m 1 n If 1 2, then m || 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 ||. m 1 2 If 1 suppl 2, then n m || n. Theorem In a plane two lines perpendicular to the same line are parallel. t If t m and t n , then m || n. m n Theorem Two lines parallel to the same line are parallel to each other If p m and m n, then p p n m n **THINK TRANSITIVE PROPERTY 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 _______________________________.” • Which pair of lines are parallel? – “Line ___ and line ____ are parallel because the _______________________________.” l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k m6=m4 l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k m 2 + m 3= m 5 l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k m 2 + m 3 + m 8 = 180 l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k 71 l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k m 1 =m 8 = 75 l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k 5 and 6 are supplementary l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k 4 and 5 are supplementary l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k 2 and 3 are complementary and m 5 = 90 l t 8 Al&t 6 j Bj&k C none 1 2 7 3 4 5 k TELL ME WHY THE LINES ARE PARALLEL? “The lines are parallel because the ________________________________________.” FILL IN THE BLANK WITH ANY OF THE FOLLOWING…. 1. Corresponding angles are congruent 2. Alternate interior angles are congruent 3. Same side interior angles are supplementary 4. 2 lines are perpendicular to the same line 5. 2 lines are parallel to the same line 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 Triangle • 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 C ▲ABC B 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 Scalene All sides congruent No congruent sides Types of Triangles (by angles) Can you name these triangles? Equiangular Acute Right Obtuse 3 acute angles 1 right angle 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 1. Label the angles of the triangle 1, 2 and 3 2. Draw a point at the tip of each vertex 3. Rip off the three angles of the triangle 1 2 3 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 1 3 2 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 mA + mB + mC = 180 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 A – Acute B – Obtuse C - Right • 47 • 43 A – Acute B – Obtuse C - Right 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 _________. Corollary 1. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. • Why? Corollary 2. Each angle of an equiangular triangle has measure___. Corollary 2. Each angle of an equiangular triangle has measure 60º. • Why? Corollary 3. In a triangle, there can be at most one _____ or _______ angle. Corollary 3. In a triangle, there can be at most one right or obtuse angle. • Why? http://www.mathsisfun.com/triangle.html Corollary 4. The acute angles of a right triangle are _____________. Corollary 4. The acute angles of a right triangle are complimentary. • Why? • 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 is an exterior angle One triangle has 6 different exterior angles m1 + m 2 = 180 3 and 4 are called remote interior angles 2 is the adjacent interior angle Measure the four angles of your triangle using the protractor. Be precise ! m1= m 2= m3= m4= • 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 m2 + m1 m2 + m3 + m4 = 180 = 180 m1 = m3 + m4 • 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” The Polygon 1. Each segment intersects exactly two other segments, one at each endpoint. 2. No two segments with a common endpoint are collinear **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 ? Not Polygons Not a segment Not Polygons Can you explain why each of these figures is NOT a polygon ? Not Polygons Doesn’t intersect 2 other segment one at each endpoint Not Polygons Can you explain why each of these figures is NOT a polygon ? 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. convex Non convex The Diagonal A segment that joins non-consecutive vertices. Choose one “start” point What polygon does not have any diagonals? # of sides Picture Name Names of Polygons Number of sides 3 4 5 6 8 10 n Name Triangle Quadrilateral Pentagon Hexagon Octagon Decagon n-gon # of sides Picture Name # of triangles 1. Choose one point on the polygon and draw all the possible diagonals from that point. 2. How many triangles do you see ? # of sides Picture Name # of Interior triangles angle sum 1. One triangle has an angle sum of 180. 2. 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 http://www.mathsisfun.com/geometry/polygons-interactive.html REGULAR POLYGON # of sides Picture Name # of Interior triangles angle sum One interior angle One exterior angle Exterior angle sum Exterior Angle + Interior Angle = 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 3 2 1 + 2 + 3 = 360 2 1 + 2 + 3 + 4 = 360 **THIS APPLIES TO ANY CONVEX POLYGON!! REGULAR POLYGONS • All the interior angles are congruent • All of the exterior angles are congruent 360 n = the measure of each exterior angle **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 A – Always B – Sometimes C – Never • The sum of the measures of the interior angles of a convex polygon is ____________ 360º A – Always B – Sometimes C – Never • 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 1. Heptagon – Find the measure of one interior and exterior angle 1. Regular Decagon White Board • An exterior angle of a regular polygon has measure 10. The polygon has _____ sides. • 36 sides White Board • An interior angle of a regular polygon has measure 160. The polygon has _____ sides. • 18 sides White Board • Three of the angles of a quadrilateral have measures 90, 60, and 115. The fourth angle has measure ___ . • 95 3.6 Inductive Reasoning Objectives • Discover the uses and hidden dangers of inductive logic • Compare inductive and deductive logic Inductive Logic Archimedes • 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. Deductive Logic • Based on accepted statements – – – – Definitions Postulates Theorems Etc… • Conclusion must be true. • Two-column proofs are deductive. 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. A – Inductive Reasoning B – Deductive Reasoning • 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. A – Inductive Reasoning B – Deductive Reasoning • 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 o 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 o **NEED TO BE ABLE TO DISTINGUISH WHICH TWO LINES ARE PARALLEL AND WHICH LINE IS TRANSVERSAL – HOW IS EACH ANGLE BEING FORMED? o 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