Unit 3 Parallel And Perpendicular Lines - Weebly

Identify Pairs of Lines andAngles“LIFE IS WHAT YOU MAKE IT.” –MR. H’SDAD

Goal Students will learn the different ways lines caninteract with each other. This included coplanar andnon-coplanar lines. Students will learn what a transversal is and thedifferent types of angles formed by a transversal.

Parallel and Skew Parallel lines-never intersect and are coplanar.Skew lines-never intersect and are notcoplanar.Parallel planes-planes that never intersectand are always the same distance apart.Perpendicular lines-lines that intersect at anright angle.lllk and n mmlk

Postulates Parallel Postulate If there is a line and a point not on the line, then there isexactly 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 isexactly one line through the point perpendicular to the givenline.

Questions What are the three different names for theinteraction of lines? Parallel, intersection, or skew. If I have a point and a line, how many lines could Idraw through the point that would be parallel to mygiven line? Exactly one T/F Two lines in a plane either intersect or areparallel.

Transversals A transversal is a line that intersects two ormore coplanar lines at different points.No transversaltransversal

Corresponding Angles Angles that are on the same side ofthe transversal and in the samecorresponding positions.2413

Alternate Interior Angles Angles that are on the opposite sidesof the transversal and between thetwo lines.42Interior13

Alternate Exterior Angles Angles that are on the opposite sidesof the transversal and outside the twolines.14ExteriorInterior32Exterior

Consecutive interior Angles Angles that are on the same side ofthe transversal and between the twolines.Exterior14Interior23Exterior

Summary Students will learn the different ways lines caninteract with each other. This included coplanar andnon-coplanar lines. Students will learn what a transversal is and thedifferent types of angles formed by a transversal.

Example 1 Which lines or planes matchthe given description. Line(s) parallel to line AD andcontaining point C DBCFHGLine(s) skew to line EF andcontaining point H CALine(s) perpendicular to line HDand containing point G BHDEGH

Example 2 Name a pair of parallel lines. Line AC ll Line BD Name a pair ofDperpendicular lines. Line AC Line BC Is Line AC Line AD?Explain. BNot enough info. It is not markedand no way to determine (yet). Please look at Ex. 2 part c inbook.AC

Example 3 Identify allcorresponding angles,alternate interiorangles, alternateexterior angles, andconsecutive interiorangles.125 6347 8

Example 4Determine what eachangle pair is called. Angle 1 and angle 2 Angle 9 and angle 13 Angle 11 and angle 2 Angle 9 and angle 4 Angle 9 and angle 16 Angle 8 and angle 14 Angle 6 and angle 7 Angle 5 and angle 1125 6347 891011 1213 1415 16

Use Parallel Lines andTransversals“THE GREAT USE OF LIFE IS TO SPEND ITFOR SOMETHING THAT WILL OUTLAST IT.” –WILLIAM JAMES

Goal Students will learn how lines being parallel affectsthe angles formed by a transversal. Students will be able to justify why angles arecongruent or supplementary based on their positionin a diagram.

Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 5m 2 621 3 7n34 4 85768p

Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, thenthe pairs of alternate interior angles are congruent. 3 6m 4 52135768n4p

Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, thenthe pairs of alternate exterior angles are congruent. 1 8m 2 72135768n4p

Consecutive interior Angles Theorem If two parallel lines are cut by a transversal, then thepairs of consecutive interior angles aresupplementary. 3 𝑎𝑛𝑑 5 𝑎𝑟𝑒 �� 4 𝑎𝑛𝑑 6 𝑎𝑟𝑒 ��m2135768n4p

Summary At this point, you should be able to: Identify special properties of the four new angle pairs whentwo parallel lines are cut by a transversal.

Example 1 Find the unknown angle measures. m 2 105 m 5 105 m 6 105 1105 2 3 m 3 m 5 180 m 3 75 m 1 75 m 4 75 m 7 75 5467

Example 2 Find the value of x. m 4 85 m 4 2x 5 180 85 2x 5 180 2x 90 180 2x 90 x 4585 4(2x 5)

Example 3 Find the value of x. 35 m 2 180 m 2 145 m 2 (7x 5) 35 2 145 7x 5 140 7x 20 x(7x 5)

Example 4 Find the value of y. 3y 87 180 3y 93 y 31(3y) 87

Example 5 Prove the Alternate Exterior Angles Theorem. Given: mlln Prove: 1 3Statements1mReasons1. mlln1. Given2. 1 22. Corresponding anglesPostulate3. 3 23. Vertical AnglesCongruence Theorem4. 1 34. Transitive Property23n

Prove Lines are Parallel“ONLY THE PERSON WHO HAS FAITH INHIMSELF IS ABLE TO BE FAITHFUL TOOTHERS.”–ERICH FROMM

Goal Students will learn how the angles formed by atransversal can be used to determine that lines areparallel. Students will be able to justify why lines are parallel.

Corresponding Angles ConversePostulate If two lines are cut by a transversal and the pairs of corresponding angles are congruent, then the linesare parallel. 1 5m21 2 6n34 3 7 4 85768p

Alternate Interior Angles Converse Theorem If two lines are cut by a transversal and the pairs ofalternate interior angles are congruent, then thelines are parallel. 3 6m21 4 5n357684p

Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal and the pairs ofAlternate Exterior Angles are congruent, then thelines are parallel. 1 8m21 2 7n357684p

Consecutive Interior Angles Converse Theorem If two lines are cut by a transversal and the pairs ofconsecutive interior angles are supplementary, thenthe lines are parallel. 3 𝑎𝑛𝑑 5 𝑎𝑟𝑒 �� 4 𝑎𝑛𝑑 6 𝑎𝑟𝑒 ��m2135768n4p

Transitive Property of Parallel Lines If two lines are parallel to the same line, then theyare parallel to each other. If mlln and nllp, then mllp.mnp

Summary At this point, you should be able to: Identify special properties of the four new angle pairs formedwhen two lines are cut by a transversal that cause lines to beparallel.

Example 1 Find the value of x that makes mlln. 105 3x 35 x105 (3x) mn

Example 2 Find the value of x that makes mlln. 5x 3x 60 2x 60(5x) m x 30(3x 60) n

Example 3 Find the value of x that makes mlln. 2x x 30 180 3x 30 180 3x 150m(2x) x 50(x 30) n

Example 4 Find the value of x that makes mlln. 2x x 30 x 30(2x) m(x 30) n(7x) pq

Example 5For each problem determine if there is enoughinformation to state that pllq.1)2)85 85 95 85 3)4)

Example 6 Prove the Alternate Interior Angles ConverseTheorem. Given: 1 2 Prove: mllnStatements1. 1 2m1Reasons1. Given2. 3 22. Vertical AnglesCongruence Theorem3. 3 13. Transitive Property4. mlln4. CorrespondingAngles ConversePostulate23n

Example 6 Prove the Alternate Interior Angles ConverseTheorem. Use a Paragraph Proof. It is given that 1 2. From the diagram 3 2due to the vertical angles congruence theorem. 3 1 due to the transitive property. Therefore mllnby the corresponding angles converse postulate.StatementsReasons1. 1 21. Given2. 3 22. Vertical Angles Congruence Theorem3. 3 13. Transitive Property4. mlln4. Corresponding Angles Converse Postulate

Find and Use Slopes of Lines“A HERO IS NO BRAVER THAN AN ORDINARYMAN (OR WOMAN), BUT HE (/SHE) IS BRAVEFIVE MINUTES LONGER.” –RALPH WALDOEMERSON

GoalYou will learn how to find the slope of the line and howthe slope of parallel and perpendicular lines relate.

Slope of a line Slope can be thought of as the steepness of a line. Slope of a nonvertical line is the ratio of verticalchange (rise) to horizontal change (run) between anytwo points on the line. 𝑚 𝑟𝑖𝑠𝑒𝑟𝑢𝑛 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 𝑦2 𝑦1𝑥2 𝑥1

Types of slope m has negative slope n has positive slopemq p has zero slope q has undefined slopenp

Slopes of Parallel Lines Two nonvertical linesare parallel iff theyhave the same slope.q Any two vertical linesare parallelmnp

Slopes of Perpendicular Lines Two nonvertical linesare perpendicular iffthe product of theirslope is -1.qmp Horizontal andvertical lines areperpendicular.n

Summary You should be able to describe slope and identifyslope of parallel and perpendicular lines.

Example 1 Find the slope of lines m, p, and n.qnmp

Example 2 Find the slope of lines m, p, and n. Determine if anyof the lines are parallel.qnmp

Example 3 Find the slope of a line perpendicular to the linecontaining the points (5, -4) and (7, 0).

Example 4 A skydiver made jumps with 3 different parachutes.The graph of his jumps are below. Which statementis true?A) Dive 2 and Dive 3 startedHeight of Eachat the same height.3000B) Dive 1 and Dive 2 lasted2500the same amount of time.2000C) Dive 1 and Dive 3 wereHeight1500the same type of(m)1000parachute.500D) Dive 2 had the parachute0that had the slowest rate-327of decent.Time (s)DiveDive 1Dive 2Dive 312

Write and Graph Equations ofLines“CERTAIN SIGNS PRECEDE CERTAINEVENTS.”–CICERO

Goal Students will be able to write and graph equations oflines.

Slope-intercept Form y mx b M is the slope of the line. B is the y-intercept of the line (where the line crossesthe y-axis). Y is y-coordinate of a point on the line. X is the x-coordinate of a point on the line. To write equation of line we need to find both b andm before writing the equation of the line.

1𝑦 𝑥 32

Standard Form Ax By C𝐶 X-intercept (where the graph crosses the x-axis) is .𝐴𝐶 Y-intercept (where the graph crosses the y-axis) is .𝐵

𝑥 ( 2)𝑦 6

Summary You should be able to identify the different ways towrite an equation of a line.

Example 1 Write the equation of each line in slope-interceptform.nmp

Example 2 Graph the equation of the line.1) y 2x 72) 5x 10y 203) y -3x4) y 85) x 1

Example 3 Write an equation of the line passing through points(2, 4) and is perpendicular to the line with theequation y 2x 7.

Example 4 What is the slope and y-intercept of these lines.1) 5x-10y -201)-10y -5x-202)𝑦 1𝑥2 22) 3x y 61)y -3x 6

Example 5 Write the equation of the line with the giveninformation.1) m 4, b -2y 4x-21) m -1, b 7y -x 7

Prove Theorems aboutPerpendicular Lines“ANXIETY IS FEAR OF ONE’S SELF.” –WILHELM STEKEL

Goal You will learn how to prove statements about paralleland perpendicular lines.

Theorem 3.8 If two lines intersect to form a linear pair ofcongruent angles, then the lines are perpendicular.If, then.

Theorem 3.9 If two lines are perpendicular, then they intersect toform four right angles.ab

Theorem 3.10 If two sides of two adjacent acute angles areperpendicular, then the angles are complementary.If12, then 1 and 2 are complementary.

Perpendicular Transversal Theorem If a transversal is perpendicular to one of twoparallel lines, then it is perpendicular to the other.If, then.

Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the sameline, then they are parallel to each other.If, then.

Distance from a line. Distance from a point to a line is the length of theperpendicular segment from the point to the line.

Summary You should be able to prove statements parallel andperpendicular lines. You should be able to determine what the distancefrom a point to a line is.

Example 1 What is the value of x?52 x 745 x52 (x 7)

Example 2 Determine which lines, if any, must be parallel in thediagram. Explain your reasoning.mnpqr

Example 3 Prove that if two sides of two adjacent angles areperpendicular, then the angles are complementary. Given: ray BA ray BCA Prove: 1 and 2 are complementaryStatementsReasons1. Ray BA Ray BC1. Given2. ABC is a right angle2. form 4 right angles3. m ABC 90 3. Definition of a right angle4. m ABC m 1 m 24. Angle Addition Postulate5. 90 m 1 m 25. Transitive Property6. 1 and 2 arecomplementary6. Definition ofcomplementary angles1B2C

Example 4 How far apart are linesm and n? Perpendicular slopeshave a product of -1m1𝑦 2 𝑥 5n1 (- )( ) -12 ( ) 2 Use distance formula1𝑦 𝑥2