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Geometry Chapter 3 Notes Name: Notes #12: Section 3.1 and Algebra Review Transversal: a line that intersects two or more coplanar lines in different points Draw: Example: Special Angles: The angles formed by lines and their transversals are special: Alternate Interior Angles: ( ) Same-side Interior Angles: ( 2 1 ) 2 6 7 8 Vertical Angles: (reminder) ( ) 2 1 3 4 6 8 6 5 7 3 5 3 4 2 4 6 5 7 ) 2 1 Same-side Exterior Angles: ( ) 1 Alternate Exterior Angles: ( 3 8 7 8 Corresponding Angles: ( 5 6 5 7 8 4 3 4 6 5 1 2 1 3 4 ) 7 8 Practice: Classify each pair of angles as alt. int., s-s int., corr, alt. ext., s-s ext, or vertical. 1.) 1, 5 2.) 2, 8 3.) 4, 2 5.) 1, 7 4.) 3, 8 6.) 4, 7 1 4 3 2 5 6 8 7 -1- Identifying Lines and Transversals: Name the two lines and the transversal that form each pair of angles. What type of special angles are they? (Hint: Trace the angles in two different colors – where they overlap is the transversal, the leftovers are the two lines.) 7.) 4, 2 lines: ____, _____ transversal: ______ type: _______________ C B 1 2 8.) B, BAD lines: ____, _____ transversal: ______ type: _______________ 4 5 3 A 9.) BAD, 5 lines: ____, _____ transversal: ______ type: _______________ D 10.) BCD, 5 E lines: ____, _____ transversal: ______ type: _______________ Algebra Practice: Solving Linear Systems by Addition/Subtraction/Elimination - Rearrange each equation so that the variable expressions are on the left side of the equals sign and the constant is on the right side of the equal sign (called Standard Form) - Multiply whole equations so that one variable expression is equal but has the opposite sign. -Add the equations together; watch one variable cancel out -Solve for BOTH variables -Write your answer as a point ( x, y ) (in alphabetical order) 11.) 2x – 3y = 8 4x + 3y = -2 12.) 13.) 3x – 5y = -11 2x – 4y = -9 14.) x = -4y - 3 3x – 2y = 5 2x + 3y – 10 = x + y - 14 x – 2y + 5 = 2x – y + 6 Solve for x and y: 15.) 70 x - 15 3x - 35 2x - y -2- Notes #13: Review of Special Pairs of Angles and Linear Systems A. Special Pairs of Angles: Name the two lines and the transversal that form each pair of angles. What type of special angles are they? Y Z 1.) 4, 3 lines: ____, _____ 5 4 6 transversal: ______ type: _______________ 2.) X , ZWV lines: ____, _____ transversal: ______ type: _______________ 3 X 3.) 6, 2 lines: ____, _____ transversal: ______ type: _______________ 4.) 4, 5 2 W 1 V lines: ____, _____ transversal: ______ type: _______________ B. Linear Systems: Substitution Method Choose one equation - get _____ ____________ alone Substitute (______________) this variable with the new expression in the second equation – USE PARENTHESES!! Solve Substitute back in the first equation to solve for the other variable Express your answer as a point (____, ____) in alphabetical order Solve using the substitution method: 5.) y = -3 6.) b=3–a 2x + 3y = 3 7.) y = -x + 15 4x + 3y = 38 b=a–1 8.) 2m – 3n = 4 m + 4n = -9 -3- Possible “strange” answers: solve using either substitution or elimination. y 2 x 1 y 3x 4 9.) 10.) y 2 x 3 12 x 4 y 16 11.) 2x – 8y = 6 x – 4y = 8 12.) y + 2x = 4 2x + 3y = 12 C. Linear Systems with fractions and decimals. To clear decimals: - multiply both sides of the equation by a multiple of 10; scoot the decimal over To clear fractions: - multiply both sides of the equation by the common denominator; cross cancel 13.) 1 1 x y 4 4 3 3x y 30 14.) 2.4 x 0.3 y 3 4x 5 y 6 -4- 0.2m 0.5n 1.4 15.) 1 1 1 m n 2 3 3 0.2m 1.2n 8.8 16.) 1 1 1 m n 4 6 6 D. Word Problems in 2 variables Write a let statement (often x = 1st number, y = 2nd number) Translate ________ _______________ into an equation Translate ________ ______________ into an equation Solve the system of equations using the substitution method Write let statements, translate to two equations, and solve using substitution: 17.) The sum of two numbers is 27. The difference of these two numbers is 3. Find the numbers. 18.) The sum of two numbers is 82. One number is twelve more than the other. Find the numbers. -5- Notes #14: Sections 3.1 and 3.2 (A) Parallel Lines Theorems & the Converse Theorems Alternate Interior Angles Theorem Converse of Alternate Interior Angles Theorem If two ________________ lines are cut by a __________, then alternate interior angles are _________________. If two lines cut by a ________________ form ______________________, then the lines are ________________. 2 1 3 4 l 2 1 6 5 8 7 3 4 6 5 m l 8 7 m Corresponding Angles Postulate Converse of the Corresponding Angles Postulate If two ________________ lines are cut by a __________, then corresponding angles are _________________. If two lines cut by a ________________ form ________________________, then the lines are ________________. 2 1 3 4 l 8 7 If two ________________ lines are cut by a __________, then same-side interior angles are _________________. 2 3 4 5 l 8 7 8 7 m m Converse of the Same-Side Interior Angles Theorem If two lines cut by a ________________ form __________________________, then the lines are ________________. 2 1 3 4 6 l 6 5 m Same-Side Interior Angles Theorem 3 4 6 5 1 2 1 5 l 6 8 7 m -6- Alternate Exterior Angles Theorem If two ________________ lines are cut by a __________, then alternate exterior angles are _________________. 2 1 3 4 l Converse of the Alternate Exterior Angles Theorem If two lines cut by a ________________ form __________________________ then the lines are ________________. 8 7 3 4 6 5 2 1 m 6 5 8 7 Same-Side Exterior Angles Theorem If two ________________ lines are cut by a __________, then same-side exterior angles are _________________. 2 1 3 4 5 l m Converse of the Same-Side Exterior Angles Theorem If two lines cut by a ________________ form _________________________, then the lines are ________________. 2 1 3 4 6 8 7 l m 5 l 6 8 7 m Complete the sentences and solve for x. 1.) The labeled angles are ____________ angles and their measures are ___________ because of the _________________________________________ 2.) The labeled angles are ____________ angles and their measures are ______________________ because of the _____________________________ 3x + 10 3x 120 100 -7- 3.) The labeled angles are ____________ angles and their measures are ___________________ because of the _____________________________ 4.) The labeled angles are ____________ angles and their measures are ___________________ because of the _____________________________ 2x + 21 7x - 4 2x + 52 4x - 8 B. Identifying Parallel Lines: Use the given information to name the lines that must be parallel. (Trace angles and look for special pairs of angles and special relationships.) 5.) 1 4 Type of angle pair: S Relationship: T 4 3 5 Parallel lines?: 2 6.) m1 m2 m3 180 Type of angle pair: Relationship: X 10 6 1 7 U 8 Parallel lines?: 9 7.) 9 2 Type of angle pair: W V Relationship: Parallel lines?: 8.) 4 7 Type of angle pair: Relationship: 9.) 2 10 Type of angle pair: Relationship: Parallel lines?: Parallel lines?: C. Special Pairs of Angles Solve for all variables. All measurements are in degrees. (Hint: extend the parallel lines and look for special pairs of angles) 10.) 11.) y x 2y 120 120 80 4x -8- 12.) 13.) 56 a d c 60 27 110 2x - 3y 2x - 5y e b 130 D. Proofs with Parallel Lines: 14.) Prove the alternate exterior angles theorem: 1 If a transversal intersects two parallel lines, then alternate exterior angles are congruent. 2 3 Given: k l Prove: 1 3 Statements Reasons 1.) 1.) 2.) 2.) Corresponding Angle Postulate 3.) 3.) 3 ____ 4.) 4.) 15.) Prove the converse of the alternate exterior angles theorem: 1 2 If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. Given: 1 3 Prove: m n 3 Statements Reasons 1.) 1.) 2.) 1 _____ 2.) 3.) 3.) Substitution 4.) 4.) Converse of the _____________________ -9- Notes #15: Sections 3.3 and 3.4 A. Parallel and Perpendicular Lines Prove the converse of the same-side interior angles theorem: If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. 1 2 3 Given: 2 and 3 are supplementary Prove: m n Statements 1.) 1.) 2.) m2 m3 _____ 2.) Definition of __________________ angles 3.) m2 m ____ _____ 3.) 4.) 4.) Substitution 5.) m2 m2 Reasons 5.) 6.) 6.) Subtraction 7.) 7.) Converse of the ________________________ If two lines are parallel to the same line, then they are ___________ to each other. In a plane, if two lines are perpendicular to the same line, then they are _____________ to each other. In a plane, if a line is perpendicular to one of two parallel lines, then it is also _____________ to the other. - 10 - Examples: For #1-3, consider coplanar lines j, k, l, and m. Given each of the following statements, what more, if anything, can you conclude about the lines? 1.) j k , k m 2.) j k , l k 3.) j k , k l , l m B. Classifying Triangles Triangles are described based on the lengths of their sides and the measures of their angles Names Based on Side Lengths Scalene Isosceles Equilateral Names Based on Angle Measures Acute Obtuse Right Equiangular Examples: For #5-7, classify each triangle (drawn to scale) by its angles and sides. 5.) 6.) 7.) For #8-10, draw a triangle, if possible, to fit each description. 8.) obtuse scalene 9.) acute isosceles 10.) right equilateral - 11 - 11.) The perimeter of ∆ABC is 32m. AB = 4x – 2, BC = 3x + 1, AC = 2x + 6. Write an equation and solve for x. Then, classify the triangle as scalene, isosceles, or equilateral. (Hint: draw a picture first and label what you know) C. The angles of a triangle: ** The sum of the interior angles of a triangle is always ______** 12.) 13.) Find the missing values and classify 2 M 4x + 11 1 3 2x + 8 m1 m2 m3 _____ L x N LMN : x ____ mL ____ mM ____ mN ____ LMN is __________ and __________ 14.) Find the values of each variable and the measure of each angle. Then classify each triangle by its angles. (All measurements shown are in degrees) w ____ A B w x ABC is _________ x ____ 61 ABD is _________ y ____ 134 z DBC is _________ z ____ D y C - 12 - Notes #16: Sections 3.4 and 5.5 A. Triangle Exterior Angle Theorem (Section 3.4) ** The measure of an exterior angle of a triangle equals the sum of the measures of the two _________________ __________________ angles.** Explore: 85 40 1.) Complete: 2 7 m1 m2 _____ 1 m2 m7 _____ m7 m3 _____ 3 6 m4 m3 m5 m6 _____ 4 m2 m3 m7 _____ 5 Diagram for #1 3.) m2 103 , m3 156 , m1 _____ 2.) Solve for x and y: 95 2 y 50 1 x 3 B. Inequalities in Triangles: The largest angle of a triangle is opposite the ________________ side of the triangle. The smallest angle of a triangle is opposite the _______________ side of the triangle. List the angles of the triangle in order from smallest to largest: (not necessarily drawn to scale) 4.) 5.) B a+1 X 11 Y 12 a a-1 A C 10 Z - 13 - List the sides of the triangle in order from shortest to longest: (not necessarily drawn to scale) 6.) 7.) X B A 46 42 Y 63 27 C Z Name the longest side: (not necessarily drawn to scale) 8.) 9.) B S 48 50 A 55 T R 72 42 C 52 65 60 U Longest: ________ D Longest: ________ The Triangle Inequality The sum of the lengths of _____________________ of a triangle is greater than the length of the _______________________. When given three side lengths of a possible triangle, they can form a triangle if: (short side) + (short side) > (long side) Is it possible for a triangle to have sides with length: (Yes/No) 10.) 8, 7, 6 11.) 9, 9, 19 12.) 2, 3, 4 When given two side lengths of a triangle, the third side of the triangle must be between their difference and their sum. (length – length) < third side < (length + length) The lengths of two sides of a triangle are given. The length of the third side must be greater than ____ but less than ____. 13.) 3, 7 14.) 12, 19 15.) 3a, 4a + 2 - 14 - Notes #17: Section 3.5 A. Polygons: ( _________ sided figures) Convex Polygon Concave Polygon Explore: Triangle Quadrilateral # of Sides = ______ # of Triangles =________ Sum of Interior Angles = ______ Sum of Exterior Angles = ______ # of Sides = ______ # of Triangles =________ Sum of Interior Angles = ______ Sum of Exterior Angles = ______ Pentagon Hexagon # of Sides = ______ # of Triangles =________ Sum of Interior Angles = ______ Sum of Exterior Angles = ______ # of Sides = ______ # of Triangles =________ Sum of Interior Angles = ______ Sum of Exterior Angles = ______ Other Common Polygons: Polygon Name # Sides Octagon Nonagon Decagon Dodecagon 18-gon 20-gon n-gon # Triangles Sum of Int. Angles Sum of Ext. Angles - 15 - Patterns for polygonal angle sums: Sum of Interior Angles Sum of Exterior Angles each interior angle + each exterior angle = ___________ Find the sum of the measures of the interior angles of each polygon: 1.) 7-gon 2.) 16-gon 3.) 22-gon Find the missing angle measures: (all measures shown are in degrees) 4.) 5.) b 130 131 a 107 85 28 160 114 123 B. Regular Polygons (where n is the number of sides in the polygon) Regular Polygons: all sides ________________ all angles ________________ Find the sum of the exterior angles for each regular polygon & then find the measure of each exterior angle in the regular polygon. 6.) Regular Pentagon 7.) Regular Hexagon 8.) Regular Octagon 9.) Regular Decagon 10.) Regular Dodecagon 12.) Regular 15-gon - 16 - Notes #18—Section 3.5 Continued Regular Polygon Formulas: Interior Angles Sum of Interior Angles = Exterior Angles Sum of Exterior Angles = Each Exterior Angle = (each interior angle) + (each exterior angle) = ________ Examples: 1) Find all of the following values for a regular dodecagon (12 sides): Number of sides = 2.) Find the following information for a regular octagon: Number of sides = Sum of exterior angles = Sum of exterior angles = Each exterior angle = Each exterior angle = Each interior angle = Each interior angle = Sum of interior angles = Sum of interior angles = Find the following information for the regular polygon: 3.) 4.) Number of sides = Number of sides = Sum of exterior angles = Sum of exterior angles = Each exterior angle = 120 Each exterior angle = 72 Each interior angle = Each interior angle = Sum of interior angles = Sum of interior angles = 5.) Number of sides = 6.) Number of sides = Sum of exterior angles = Sum of exterior angles = Each exterior angle = Each exterior angle = Each interior angle = 144 Each interior angle = Sum of interior angles = Sum of interior angles = 2340 - 17 - Complete the chart for the regular polygons: 7. 8. # of sides (n) 6 9. Sum of Exterior Angles Each Exterior Angle 90˚ Each Interior Angle 2880˚ Sum of Interior Angles 10.) Given: 1 3 Prove: m n 1 2 3 Statements Reasons 1.) 1.) 2.) 2.) 3.) 3.) 4.) 4.) - 18 - Notes #19: Quiz Review 1.) Solve for x and y: 2.) a b, c d Solve for w, x, y, and z. c 3x 5 y 1 d a 4 x 2 y 36 y x 112 b 27 z w w = ____, x = ____, y = ____, z = ____ 3.) m n . Solve for x and y: 4.) The perimeter of ABC is 82ft. Solve for x and y and classify the triangle based on its sides and angles. B m 5y + 35 10x - 20 7y + 9 60 n x- 3 A x+ 6 3y - 1 5y - 23 C 3x - 11 x = ___, y = ____, classify: _________, _________ 5.) Solve for x and y. 6.) Name the sides in order from smallest to largest: B 4x + 13 72 65 A 2y + 15 5x 28 70 C x + 17 80 45 x = _____, y = _____ D 7.) Complete for the regular decagon: 8.) Complete for the regular polygon: Number of sides = Number of sides = Sum of exterior angles = Sum of exterior angles = Each exterior angle = Each exterior angle = Each interior angle = Each interior angle = 120 Sum of interior angles = Sum of interior angles = - 19 - Notes #20: Section 3.6 A. Slope: Slope is used to describe the ________________ and _________________ of lines. Sketch a line with: a) positive slope b) negative slope c) zero slope d) undefined slope A line is shown. Use two marked points and count “rise over run” to find the slope of the line. 1.) 2.) 3.) Slope = Slope = Slope = Without using a graph and given two points: x1, y1 and x2 , y2 Slope = m = 0 0 n y2 y1 x2 x1 n undefined 0 For #4-5, find the slope of the line passing through the two given points: 4.) (-4, 1), (3, 2) 5.) (6, -3) and (2, -1) 6.) A line with slope 7 passes through the 3 points (1, 2) and (-2, y). Find y. - 20 - B. Graphing Lines There are many ways to graph a line. You need to know how to graph a line: (i) given a point and a slope (ii) by finding the y-intercept and the slope of the line (i) Graphing lines using a point and a slope Point P which lies on the line and the slope of the line are given. Sketch the line and find the coordinates of two other points on the line 7.) P (-2, 1); slope = 4 5 1st point: 8.) P (0, -3); slope = -2 1st point: 2nd point: 2nd point: 9.) P (2, 0); slope = 2 3 1st point: 2nd point: - 21 - (ii) Graphing Lines using the slope and y-intercept: - Get y alone so the equation is in y = mx + b form (m = _________, b = _________) - Graph b first. This point goes on the ____ axis. - Use slope and count rise over run to the next point(s). When you have at least three points, then connect the points to make a line. - Label your graphed line with the original equation Most common errors: Graphing b on the x-axis instead of the y-axis Graphing the slope in the wrong direction (e.g. forgetting a negative) 10.) 1 y x5 2 ( I’m already in slope-intercept form!) m = ___ ( graph me second! Watch the negative!) b = ___ ( graph me first! I go on the y-axis!) 11.) x – 2y =2 ( Get me in slope-intercept form first) m = ______ b = ______ - 22 - 12.) x + 3y = -6 ( Get me in slope-intercept form first) m = ______ b = ______ Special Cases: Graphing Horizontal and Vertical Lines 13.) x = 4 (This equation describes the line for which ALL points have an x-coordinate of 4. There are no restrictions on the value of y). 14.) y = -2 (This equation describes the line for which ALL points have an y-coordinate of -2. There are no restrictions on the value of x). Use the pattern you found above to complete these sentences: • Any line in the form x = _____ is a ________________ line because it intersects the ___ _________ • Any line in the form y = _____ is a ________________ line because it intersects the ___ _________ - 23 - Use this pattern to graph these lines without a table of solutions. 15.) y = 3 16.) x = -2 17.) y = -4 - 24 - Notes #21: Writing Linear Equations A. Converting equations of lines: Lines can be written in either Slope-Intercept form (y = mx + b) or Standard Form (Ax + By = C). You need to know how to convert from one to the other. Converting to Slope-Intercept Form Converting to Standard Form Goal: y = mx + b (where m and b are integers or fractions) Goal: Ax + By = C (where A, B, and C are integers and where A is positive) Get y alone Reduce all fractions 1.) Convert to slope-intercept form: 4x – 12y = 8 Get x and y terms on the left side and the constant term on the right side of the equation Multiply ALL terms by the common denominator to eliminate the fractions If necessary, change ALL signs so that the x term is positive 2.) Convert to standard form: y 2 x 5 3 B. Writing linear equations given the slope and y-intercept - Find the slope (m) and y-intercept (b) [If the given information is a graph, then you will have to count by hand to find these values.] - Fill in m and b so you have an equation of the line in y = mx + b form. y = ________ x + ____________ ( Put m here!) ( Put b here!) 3.) Find the equation of the line with slope of 5 and y-intercept of -2. Write in standard form. 4.) Find the equation of the given line in slope-intercept form. 5.) Write the equation of a line that has the same slope 4 as y x 3 and has a y-intercept 5 of 1. Write in standard form. - 25 - C. Writing linear equations given the slope and a point plug slope = m into y = mx + b name your point (x, y) and plug these values in for x and y solve for b plug m and b back into y = mx + b convert to standard form, if necessary ** Remember to leave x and y as variables! ** 6.) Find the equation of the line in slope-intercept form with slope of -2 and going through (-1, 3) in. 7.) Find the equation of the line in standard form with 1 slope of and going through (6, -2). 3 8.) Find the equation of the line in slope-intercept 2 form with slope and passing through the point 5 (-3, 7). 9.) Find the equation of the line in standard form with 3 slope and going through (-8, 6). 4 - 26 - Notes 22—Writing Linear Equations Continued Writing linear equations given two points find the slope pick one of your points to be x and y plug m, x, y into y = mx + b solve for b; plug m and b into y = mx + b convert to standard form, if necessary ** Remember to leave x and y as variables! ** 1.) Find the equation of the line in slope-intercept form going through (-3, 1) and (4, 8). 2.) Find the equation of the line in standard form with x-intercept 3 and y-intercept -2. 3.) Find the equation of the line going through (5, 2) and (-1, 3) in standard form. 4.) Find the equation of the line with x-intercept 5 and y-intercept -4 in slope-intercept form. 5.) Find the equation of the line in standard form going through (-8, -2) and (-6, 4). 6.) Find the equation of the line in standard form with x-intercept -8 and y-intercept -2. - 27 - 7.) Find the equation of the line in slope-intercept 1 form with slope and passing through the point 4 (-12, 3). 8.) Find the equation of the line in standard form with 8 slope and passing through the point (27, 5) 9 - 28 - Notes #23: Section 3.7 A. Review - Writing Linear Equations: 1.) Find the equation of the line with x-intercept -1 and y-intercept 2 in standard form. 2.) Find the equation of the line going through (4, 3) with x-intercept 6 in standard form. B. Parallel and Perpendicular Lines For #3-4, a pair of parallel lines and a pair of perpendicular lines are graphed below. Use the graphs to find the slope of each of the four lines and to complete the sentences. 3.) Slope of l1 : Slope of l2 : Parallel lines have _________ slopes. - 29 - 4.) Slope of l3 : Slope of l4 : Perpendicular lines have ____________, _____________ slopes. The slope of a line is given. Find the slope of a line parallel to it and the slope of a line perpendicular to it: 2 5.) m = 6.) m = 7 7.) m = 0 3 Parallel: Parallel: Parallel: Perp.: Perp.: Perp.: Are the lines with these slopes parallel, perpendicular, or neither? 2, 4 3 6 8.) 9.) -4, 4 10.) -1, 1 11.) Find the slope of a line parallel and perpendicular to AB where A(-3, 1) and B (2, 4) Parallel:_______ Perp:______ For #12-14, state whether the given pair of lines is parallel, perpendicular, or neither: 12.) 3 y x 1 4 8 x 6 y 12 13.) 1 y x5 2 2x 4 y 9 14.) y 5 x 4 x 5 y 4 - 30 - C. Writing linear equations given a point and another line (parallel or perpendicular to your line) find m from the given line if the line is parallel, this is your m; if the line is perpendicular, find its ______________ _______________ plug m, x, y into y = mx + b solve for b; plug m and b into y = mx + b convert to standard form, if necessary ** Remember to leave x and y as variables! ** 15.) Find the equation of the line going through 16.) Find the equation of the line going through (1, 2) and parallel to y = 3x + 4 in slope-intercept (3, -2) and perpendicular to x – 4y = 3 in standard form. form. 17.) Find the equation of the line going through (-1, 5) and perpendicular to y = 3x + 4 in slopeintercept form. 18.) Find the equation of the line going through (9, -3) and parallel to 2x – 3y = 3 in standard form. - 31 - Notes #24: Review You can now solve linear systems (a set of 2 lines) using algebra (substitution/elimination) AND using coordinate Geometry (graphing). You should get the same answer for both methods. - solve the equations using substitution or elimination; write your answer as a point ( , ) - graph the two lines using either the intercept method OR slope-intercept method - confirm that the two lines intersect (meet) at your solution point 1.) y=x–2 x+y=4 2nd Method: graphing (graph both lines on the coordinate plane below) 1st Method: substitution or elimination solution: ( 2.) , ) 2x – y = -3 x + 2y = 6 2nd Method: graphing (graph both lines on the coordinate plane below) 1st Method: substitution or elimination solution: ( , ) - 32 - Chapter 3 Study Guide For #1-4, identify whether the angles are vertical angles, same side interior angles, corresponding angles, alternate interior angles, same-side exterior angles, or alternate exterior angles. 1.) 2 and 6 2.) 1 and 6 3 7 4 3.) 5 and 2 2 4.) 4 and 7 5 1 6 For #5-6, find the slope of the line passing through the two points 5.) (-3, 2) and (4, 1) 6.) (-9, 2) and (2, 2) 7.) The slope of line l is given. Find the slope of the line parallel to it and the slope of the line perpendicular to it: a) -2 b) 3/2 Parallel:____ Parallel:___ Perp.: _____ Perp.:_____ For #8-10, name the two lines and transversal that form each pair of angles: 1 2 9.) BAD, CDA lines: ____, ____ trans: ______ 10.) BAD, 5 lines: ____, ____ trans: ______ C B 8.) 1, 3 lines: ____, ____ trans: ______ 4 5 3 A E D In the diagrams, the lines shown are parallel. Write an equation and solve for x and y. Justify your work with a postulate or theorem. 12.) 11.) 2y + 20 2x+40 2x + 10 5y+20 x+80 3y + 65 Find the values of x and y. 13.) 40 14.) 5y y 20 15x-20 x 5x - 33 - Define your variables, write an equation, and solve: 15.) The sum of two numbers is 10. The difference of the first number and twice the second number is 1. Find the numbers. In each exercise, some information is given. Use this information to name the segments that must be parallel. If there are no such segments, write none. A B 16.) 3 10 17.) 7 10 2 1 18.) 2 3 3 10 F 9 11 8 19.) 9 5 4 C 5 6 7 D E Solve for x and y: 20.) x+ y 130 120 Solve for x and y: 1 1 x y 1 2 4 21.) 2 1 x y 8 3 2 6x - 2y 22.) Prove the converse of the alt ext angles theorem: If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. Given: 1 3 Prove: m n Statements 1.) 1.) 2.) 2.) 3.) 3.) 4.) 4.) 1 2 3 Reasons - 34 - 23.) Given: l m 4 Prove: m2 m4 180 3 l 2 m 1 Statements Reasons 1.) 1.) 2.) 2.) 3.) 3.) 4.) 4.) 24.) Given: AB = CD AE = FD A Prove: EB = CF E B C F Statements Reasons 1.) 1.) 2.) 2.) 3.) 3.) 4.) AE = FD D 4.) 5.) 5.) For #25-26, classify the triangles based on their sides and their angles: 25.) 26.) 8x - 20 3x + 5 2x 60 60 - 35 - For #27-29, use the diagram for reference. Show all equations and work. 27.) If m6 42 and m8 61, then m10 ____ 11 8 6 7 10 9 If m6 7 x, m7 2 x 5, 28.) and m11 6 x + 35, then x = ___. If m8 7 x 2, m7 4 x 7, 29.) and m9 10 x + 3, then x = ___. For #30-31, a, b, c, and d are distinct coplanar lines. How are a and d related? 30.) a b, b c, c d 31.) a b, b c, c d For #32-33, find the measure of each interior angle and each exterior angle of each regular polygon. 32.) octagon 33.) pentagon Complete the table for regular polygons 34.) Number of Sides Sum of exterior angles Measure of each exterior angle Measure of each interior angle Sum of interior angles 6 20 162 - 36 - Graphing Linear Equations: 35.) Graph each line using the slope and y-intercept: 3 a) y = x + 1 b) 2x + y = 4 c) 3x – 2y = 8 2 36. Find the slope and y-intercept of each line: a) y = 3x – 5 b) y = 3 c) 3x – 2y = 4 37. Find the intersection of the two lines using the substitution or elimination method. x + 2y = 8 and 2x + 3y = 10 38. Explain why these two lines will not intersect y = 2x – 1 and 8x – 4y = 16 Writing Linear Equations: Write an equation of the line with: 39. y-intercept -2 and slope -4 in standard form 40. x-intercept 4 and y-intercept -2 in slopeintercept form - 37 - 42. through (6, 2) and parallel to x – 2y = 5 in standard form 41. through (1, -2) with slope -3 in slopeintercept form 43. through (2, -1) and perpendicular to x + 3y = 7 in standard form 44. through (3, 2) and (4, 7) in slope-intercept form 45. through (3, -2) and (7, -2) in slope-intercept form 46. through (4, -3) and with x-intercept -2 in standard form 47. x-intercept -3 and y-intercept 5 in standard form 49. through (3, -3) and perpendicular to 2x – y = 1 in slope-intercept form For #50-52, are the given lines parallel, perpendicular, or neither? 1 y x2 50.) 3 2 x 6 y 10 51.) 1 y x2 3 2 x 6 y 10 1 y x2 52.) 3 6 x 2 y 10 - 38 - For questions 53-54, can a triangle have sides with the given lengths? Write yes or no and show work to support your answer. 53. 4m, 7m, and 8m 54. 18ft, 20ft, and 40ft. 55. The lengths of two sides of a triangle are 6 km and 8 km. Describe the lengths possible for the third side. For questions 56-57, list the sides in order from shortest to longest. 56. 57. B B C 60 61 47 A 65 59 C 55 60 41 D A For questions 58-59, list the angles from smallest to greatest. 58. In ∆ABC, AB = 11, BC = 12, and AC = 10. 59. R b Q b +1 b -1 P - 39 -