Unit 3 Notes: Parallel Lines, Perpendicular Lines, And .

Unit 3 Notes: Parallel Lines, Perpendicular Lines, and Angles3-1 TransversalREVIEW:*Postulates are Fundamentals of Geometry (Basic Rules)To mark line segments as congruent draw thesame amount of tic marks on each one.CNaming the angle in 4 ways:AAA1) B2) ABC3) 14) CBA1BAA* Transversal – a line that intersects two unique lines at two different pointsThere are five types of angles with regard to a transversal*Corresponding angles: Pairs of angles that are in the same locationwith regards to the transversal 1 and 5, 3 and 7, 2 and 6, 4 and 8*Alternate Interior Angles: Pairs of angles on opposite sides of the transversalbetween the intersected lines. 3 and 6, 4 and 5*Alternate Exterior Angles: Pairs of angles on opposite sides of the transversaloutside the intersected lines. 1 and 8, 2 and 7*Consecutive Interior Angles: Pairs of angles on the same side of the transversalBetween the intersected line. aka same side interior 3 and 5, 4 and 6*Consecutive Exterior Angles: Pairs of angles on the same side of the transversaloutside the intersected lines. aka same side exterior 1 and 7, 2 and 8,1

Using Transversal aIdentify all pairs ofAlternate Interior Angles: 4& 5, 3 & 6Alternate Exterior Angles: 1 & 8, 2 & 7Consecutive Interior Angles: 3 & 5, 4& 6Consecutive Exterior Angles: 1 and 7, 2 and 8Corresponding Angles: 1& 5, 2& 6, 3& 7, 4& 8Using Transversal bIdentify all pairs ofAlternate Interior AnglesAlternate Exterior AnglesConsecutive Interior AnglesConsecutive Exterior AnglesCorresponding AnglesUsing Transversal cIdentify all pairs ofAlternate Interior AnglesAlternate Exterior AnglesConsecutive Interior AnglesConsecutive Exterior AnglesCorresponding Angles2

3-2 Properties of Perpendicular LinesPerpendicular Lines: If two lines are perpendicular, then they intersect to form four right angles.The symbol for "is perpendicular to" is Linear Perpendicular Line Theorem: If two lines intersect to form a linear pair of congruent angles,then the lines are perpendicular.g𝑔 ℎhPerpendicular Complementary Theorem: If two sides of two adjacent acute angles are perpendicular,Then the angles are complementary.Developmental Proof of Linear Perpendicular Theorem:1) 1 22)3) 1 & 2 are a linear pair4)5) 𝑚 1 𝑚 1 1806)7) 𝑚 1 908)9) 𝑔 ℎGiven: 1 2, 1 & 2 are a linear pair.Prove: 𝑔 ℎg1StatementReason1) Given2)3) Given4) Linear Pair Post5) Sub. Prop. of 6)7) Div. Prop of 8)9) Def. of Lines2h3

3-3 Properties of Parallel LinesFOLLOWING THEOREMS ARE USEFUL IN PROOFS DEALING WITH PARALLEL LINES***Corresponding Angles Postulate: - If a transversal intersects two parallel lines, then thecorresponding angles formed are congruent.If a b, then all corresponding angles are congruent 1 5, 2 6, 3 7, 4 8If m 1 52 , using the corresponding angles postulate,Vertical angles theorem, and linear pair postulate, find the measure of the other angles.m 2 m 4 m 6 m 3 m 5 m 7 m 8 *Alternate Interior Angles Theorem – If a transversal intersects two parallel lines, then theAlternate Interior Angles are congruent.If a b, then all pairs of Alternate Interior Angles are congruent. 3 6, 4 5If m 4 (3x 4) and m 5 67 , then what is the value of x?*Alternate Exterior Angles Theorem – If a transversal intersects two parallel lines, then theAlternate Exterior Angles are congruent.If a b, then all pairs of alternate exterior angles are congruent. 1 8, 2 7If m 1 32 , then what is m 8?4

*Consecutive Interior Angles Theorem – If a transversal intersects two parallel lines, thenthe Consecutive Interior Angles are supplementary.If a b, then all pairs of consecutive interior angles are supplementary.m 3 m 5 180 m 4 m 6 180 If m 4 (4x 12) and m 6 120 , then what is the value of x and m 4?*Consecutive Exterior Angles Theorem – If a transversal intersects two parallel lines, thenthe Consecutive Exterior Angles are supplementary.If a b, then all pairs of consecutive interior angles are supplementary.m 1 m 7 180 m 2 m 8 180 If m 1 (5x 10) and m 7 60 , then what is the value of x and m 1?Developmental proof of the alternate interior angles theorem Hint: Use the corr ’s PostulateStatements1) a bReasons1)2) 4 12)3) 1 53)4) 4 54)5

Examples using geometric shapes: Find the value of x.3-4 Proving Lines ParallelConverse theorems about transversals and parallel lines***Converse to the corresponding angles postulate –If two lines and a transversal formcorresponding angles that are congruent, then the lines are parallel ***If 1 5,then a bIf m 1 2x 66 and m 5 8x -24, then what is the value of x? and what is the measure ofeach angle?***Converse to the Alternate interior angles theorem -If two lines and a transversal formalternate interior angles that are congruent, then the lines are parallel.***If 2 5,then a bIf m 2 3x -4 and m 5 2x 16, then what is the value of x?. and what is the measure ofeach angle?6

***Converse to the Alternate Exterior Angles Theorem-If two lines and a transversal formalternate exterior angles that are , then the lines are ***If 3 7,then a bIf m 3 4x - 5 and m 7 2x 37, what is the value of x?***Converse to the Consecutive interior angles theorem -If two lines and a transversal formconsecutive interior angles that are , then the lines are .***If m 4 m 5 ,then If m 4 2x -4 and m 5 3x – 16, then what is the value of x?. and what is the measure ofeach angle?***Converse to the Consecutive Exterior Angles Theorem -If two lines and a transversal formconsecutive exterior angles that are , then the lines are .***If m 1 m 7 ,then 1ab7If m 1 6x - 14 and m 7 3x – 13, then what is the value of x? and what is the measure ofeach angle?7

3-5 Transversal Parallel and Perpendicular Line TheoremsTransitive Parallel Lines Theorem*** If two lines are parallel to the same line, then they are parallel to each other.***If a b and b c, then a c.Parking Lot Theorem***If two coplanar lines are perpendicular to the same line, then they are parallel to eachother.***If a d and b d, then a bDevelopmental proof of the parking lot theorem.StatementsReasons1) a d, b d1)2) 1 and 2 are right ’s2)3) 1 23)4) a b4)StatementsReasons1) a b, b c, and c d1)2) a c2)3) a d3)8

Converse to the Parking Lot Theorem***In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to theother.***If a b and a d, then b dA ladder is an excellent example of all the theorems comparing perpendiculars with parallels.What can you conclude about a ladder for each given:1) Given: The rungs are perpendicular to one side:a. Conclusion:2) Given : Each side is perpendicular to the top rung:a. Conclusion:3) Given: Each rung is parallel to the top rung:a. Conclusion:With the given information, we need to find the relationship between line a and line d. (DRAWA PICTURE)1) a b, b c, c dWhat do we know about a and d?2) a b, b c, c d What do we know about a and d?3) a b, b c, c d What do we know about a and d?Indirect ProofGiven: m 1 m 2Prove: line k is not perpendicular to line m.1k2m9

3-6 Lines in the Coordinate PlaneFinding an Equation of a LineThe Point-Slope EquationA non-vertical line with slope m and containing a point(𝑥1 , 𝑦1 ) has the point-slope equation𝑦 𝑦1 𝑚(𝑥 𝑥1 )Example 1Write an equation for the line with slope 3 that contains the point (5, 2). Express the equation inslope-intercept form.𝑦 𝑦1 𝑚(𝑥 𝑥1 )𝑦 2 3(𝑥 5)𝑦 2 3𝑥 15𝑦 3𝑥 13Try This. Write an equation for each line with the given point and slope. Express the equationin slope-intercept form.a. (3, 5), m 6b. (1, 4), m 23We can also use the point-slope equation to find an equation of a line if we know any two pointson the line.Example 2.We first find any two points on the line. Use (1, 1) and (2, 3). We next find the slope.3 1 2𝑚 22 1 110

We can now use point-slope equation to find an equation for the line. We can use either point.Using (1, 1) may make the computation easier.𝑦 𝑦1 𝑚(𝑥 𝑥1 )𝑦 1 2(𝑥 1)𝑦 1 2𝑥 2𝑦 2𝑥 1Try This. Write an equation for each line in slope-intercept form.g.h.11

3-7 Slopes of Parallel and Perpendicular LinesParallel lines are lines in the same plane that never intersect.All vertical lines are parallel.Non-vertical lines that are parallel are precisely those that have the same slope and different yintercepts.The graphs below are for the linear equations y 2x 5 and y 2x – 3.The slope of each line is 2 and the y-intercepts are 5 and -3 so these lines are parallel.Example 1.Determine whether the lines of y -3x 4 and 6x 2y -10 are parallel.We must find each equation for y.y -3x 46x 2y -102y -6x – 10y -3x – 5The graphs of these lines have the same slope and different y-intercepts. Thus they are parallel.Try This. Decide whether the graphs of the equations are parallel.1) 3x – y -5 and 5y – 15x 102) 4y -12x 16 and y 3x 4Perpendicular Lines are lines that intersect to form a 90 angle (or a right angle).A vertical line and a horizontal line are perpendicular.1Algebraically, the product of the slopes of perpendicular lines is -1. The slopes are 2 and 2have a product of -1 so these lines are perpendicular.12

Example 2Tell whether the graphs of 3y 9x 3 and 6y 2x 6 are perpendicular lines.We first solve for y in each equation to find the slopes.13y 9x 36y 2x 6𝑇ℎ𝑒 𝑠𝑙𝑜𝑝𝑒𝑠 𝑎𝑟𝑒 3 𝑎𝑛𝑑 3Y 3x 16y -2x 61Y 3x 11The products of the slopes of these lines is 3(- 3) -1. Thus the lines are perpendicular.Try This. Tell whether the graphs of the equations are perpendicular.1) 2y – x 2 and y -2x 42) 4y 3x 12 and -3x 4y – 2 0Example 3Write an equation for the line containing (1, 2) and perpendicular to the line y 3x – 1.The slope of the line y 3x – 1.1The slope of the line y 3x – 1 is 3. The negative reciprocal of 3 is - 3.𝑦 𝑦1 𝑚(𝑥 𝑥1 )1𝑦 2 (𝑥 1)311𝑦 2 𝑥 3317𝑦 𝑥 3313

Try This. Write an equation for the line containing the given point and perpendicular to thegiven line.a. (3, 2); y 2x 4b. (-1, -3); x 2y 814

3-7 Slopes of Parallel and Perpendicular Lines Parallel lines are lines in the same plane that never intersect. All vertical lines are parallel. Non-vertical lines that are parallel are precisely those that have the same slope and different y-intercepts. The graphs below are for the linear equations y 2x 5 and y 2x – 3.