Geometry Chapter 3 Notes Notes #13: Section 3.1 and Algebra Review. A. Definitions:


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1 Geometry Chapter Notes Notes #: Section. and Algebra Review A. Definitions: Parallel Lines: Draw: Transversal: a line that intersects two or more coplanar lines in different points Draw: Example: Example: Parallel Planes: Draw: Example: Practice:.) Name a plane parallel to ABGH..) Name three lines parallel to CF.) Classify the planes as intersecting or parallel: ADEH and BCFG.) Draw and label the figure described: AB and XY are coplanar and parallel. h is a transversal that intersects them at points C and Z, respectively. B. Special Angles: The angles formed by lines and their transversals are special: Alternate Interior Angles: ( ) Same side Interior Angles: ( )
2 Corresponding Angles: ( ) Alternate Exterior Angles: ( ) Same side Exterior Angles: ( ) Vertical Angles: (reminder) ( ) Practice: Classify each pair of angles as alt. int., s s int., corr, alt. ext., s s ext, or vertical..), 6.), 8 7.), 8.), ), 7 0.), 7 C. 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.).), lines:, transversal: type: B C.) B, BAD lines:, transversal: type: A D E.) BAD, lines:, transversal: type:.), BCD lines:, transversal: type:
3 D. 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).) x y = 8 6.) x = y x + y = x y = 7.) x y = 8.) x + y 0 = x + y x y = 9 x y + = x y + 6 Solve for x and y: 9.) x x 70 x y
4 Notes #: Sections. and. A. Relationships formed by parallel lines Alternate Interior Angles Theorem If two lines are cut by a, then alternate interior angles are. (Its converse): If two lines cut by a form congruent alternate interior angles, then the lines are. Corresponding Angles Postulate If two lines are cut by a, then corresponding angles are. (Its converse): If two lines cut by a form congruent corresponding angles, then the lines are. Same Side Interior Angles Theorem If two lines are cut by a, then same side interior angles are. (Its converse): If two lines cut by a form supplementary same side interior angles, then the lines are. Alternate Exterior Angles Theorem If two lines are cut by a, then alternate exterior angles are. (Its converse): If two lines cut by a form congruent alternate exterior angles, then the lines are. Same Side Exterior Angles Theorem If two lines are cut by a, then same side exterior angles are. (Its converse): If two lines cut by a form supplementary same side exterior angles, then the lines are.
5 Complete the sentences and solve for x..) The labeled angles are angles and their measures are because of the.) The labeled angles are angles and their measures are because of the 00 x + 0 x 0.) The labeled angles are angles and their measures are because of the.) The labeled angles are angles and their measures are because of the x + 7x x + x 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.).) Type of angle pair: Relationship: S T Parallel lines?: 6.) m + m + m = 80 Type of angle pair: Relationship: U Parallel lines?: 7.) 9 Type of angle pair: Relationship: 9 W V Parallel lines?: 8.) 7 Type of angle pair: Relationship: 9.) 0 Type of angle pair: Parallel lines?: Relationship: Parallel lines?:
6 6 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) 0.) y x.) 0 y 0 80 x.).) 6 c d 7 0 x y x y a b 60 e 0 D. Proofs with Parallel Lines:.) Prove the alternate exterior angles theorem: If a transversal intersects two parallel lines, then alternate exterior angles are congruent. Given: k l Prove: k l.) Statements.) Reasons.).).).) Corresponding Angle Postulate.).)
7 7.) Prove the converse of the alternate exterior angles theorem: If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. Given: Prove: m n m n.) Statements.) Reasons.).).).) Substitution.).) If two lines and a transversal form angles, then the two lines are. Notes #: Sections. and. 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. Given: and are supplementary Prove: m n m n.) Statements.) Reasons.) m + m =.) m + m =.).) m = m 6.) 7.).) Definition of angles.).) Substitution.) 6.) Subtraction 7.) If two lines and a transversal form angles, then the two lines are.
8 8 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. Examples: For #, consider coplanar lines j, k, l, and m. Given each of the following statements, what more, if anything, can you conclude about the lines?.) j k, k m.) j k, l k.) 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 Sides Scalene Isosceles Equilateral
9 Angles Acute Obtuse Right 9 Equiangular Examples: For # 7, classify each triangle (drawn to scale) by its angles and sides..) 6.) 7.) For #8 0, draw a triangle, if possible, to fit each description. 8.) obtuse scalene 9.) acute isosceles 0.) right equilateral.) The perimeter of ABC is m. AB = x, BC = x +, AC = x + 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 **
10 .) m + m + m =.) Find the missing values and classify LMN : M x = L x + 8 LMN x + x is and N 0 m L = m M = m N =.) 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 x = ABC is 6 y = ABD is z z = DBC is D y C ** The measure of an exterior angle of a triangle equals the sum of the measures of the two angles.** Explore: 8 0.) Complete: 7 6 m + m = m + m 7 = m 7 + m = m + m + m + m 6 = m + m + m 7 = Diagram for #
11 6.) Solve for x and y: 7.) m = 0, m = 6, m = 9 0 x y Notes #6: Section. and Algebra Review A. Polygons: ( sided figures) Convex Polygon Concave Polygon Explore: Triangle sides Quadrilateral sides Sum of Interior Angles = Sum of Exterior Angles = Pentagon sides Sum of Interior Angles = Sum of Exterior Angles = Hexagon sides Sum of Interior Angles = Sum of Exterior Angles = Sum of Interior Angles = Sum of Exterior Angles = Other Common Polygons: Octagon: sides Nonagon: sides Decagon: sides Dodecagon: sides 8 gon: sides 0 gon: sides n gon: sides
12 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 angles of each polygon: (interior angles).) decagon.) octagon.) gon Find the missing angle measures: (all measures shown are in degrees).).) 0 a b B. Regular Polygons (where n is the number of sides in the polygon) Regular Polygons: all sides all angles Interior Angles Sum of Interior Angles: Sum of Exterior Angles: Exterior Angles OR Each Interior Angle: Each Exterior Angle: Extra Trick: (each interior angle) + (each exterior angle) =
13 Complete the chart using these relationships. (Pictures may help!) # of sides (n) 6 8 Sum of Exterior Angles Each Exterior Angle 7 Each Interior Angle 90 Sum of Interior Angles ) 7.) 8.) 9.) 0.).) D. Word Problems: Define two variables and write two equations to solve..) The sum of two numbers is 8 and their difference is 6. Find each of the numbers..) The sum of two numbers is one more than twice the smaller number. Their difference is seven less than twice the larger number. Find the numbers.
14 D. Algebra Practice: Solving linear systems with fractions and/or decimals. To clear decimals: multiply both sides of the equation by a multiple of 0; scoot the decimal over To clear fractions: multiply both sides of the equation by the common denominator; cross cancel.) 0.m 0.n =. m + n =.) 0.m +.n = 8.8 m n = 6 6 Notes #7: Section.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
15 A line is shown. Use two marked points and count rise over run to find the slope of the line..).).) Slope = Slope = Slope = Without using a graph and given two points:, x, y ( x y ) and ( ) Slope = m = y x y x For #, find the slope of the line passing through the two given points: 0 n n 0 = 0 = undefined.) (, ), (, ).) (6, ) and (, ) 6.) A line with slope 7 passes through the points (, ) and (, y). Find y. 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 x intercept and y intercept, and (iii) by finding the y intercept and the slope of the line.
16 6 (i) Graphing lines using a point and a slope A point P on a line and the slope of the line are given. Sketch the line and find the coordinates of two other points on the line 7.) 8.) 9.) P (, ); slope = st point: nd point: P (0, ); slope = st point: nd point: P (, 0); slope = st point: nd point: (ii). Graphing Lines using the x and y intercepts. The intercepts are the point(s) where a line intersects the axes of the coordinate plane. Find the x and y intercepts (by setting the opposite variable to zero) Write these answers as two different points Graph and connect these points to graph the line Label the graphed line with the original equation Most common error: Forgetting that the intercepts are two different points and graphing as just one 0.) x + y = x intercept y intercept (set y = 0) (set x = 0) x int: (, 0) y int: (0, ).) x y = x int: (, ) y int: (, ) y x
17 7 (iii) 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).) y = x ( I m already in slope intercept form!) m = ( graph me second! Watch the negative!) b = ( graph me first! I go on the y axis!).) x y = ( Get me in slope intercept form first) m = b =.) x + y = 6 ( Get me in slope intercept form first) m = b = y x y x Graph for # AND # (be sure to label your lines!)
18 Special Cases: Graphing Horizontal and Vertical Lines 8.) x = (This equation describes the line for which ALL points have an x coordinate of. There are no restrictions on the value of y). 6.) y = (This equation describes the line for which ALL points have an y coordinate of. There are no restrictions on the value of x). 7) 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 Use this pattern to graph these lines without a table of solutions. 8.) y = 9.) x = 0.) y =
19 Notes #8: Writing Linear Equations 9 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 Goal: y = mx + b (where m and b are integers or fractions) Get y alone Reduce all fractions.) Convert to slope intercept form: x y = 8 Converting to Standard Form Goal: Ax + By = C (where A, B, and C are integers and where A is positive) 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.) Convert to standard form: y = x 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!).) Find the equation of the line with slope of and y intercept of. Write in standard form..) Find the equation of the given line in slope intercept form..) Write the equation of a line that has the same slope as y = x and has a y intercept of. Write in standard form.
20 C. Writing linear equations given the slope and a point 0 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 with slope of and going through (, ) in slope intercept form. 7.) Find the equation of the line with slope of and going through (6, ) in standard form. 8.) Find the equation of the line in slope intercept form with slope and passing through the point (, 7). D. 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 9.) Find the equation of the line going through (, ) and (, 8) in slope intercept form. ** Remember to leave x and y as variables! ** 0.) Find the equation of the line with x intercept and y intercept in standard form..) Find the equation of the line going through (, ) and (, ) in standard form..) Find the equation of the line with x intercept and y intercept in slope intercept form.
21 Notes #9: Section.7 A. Review Writing Linear Equations:.) Find the equation of the line with slope of and going through (, ) in slope intercept form..) Find the equation of the line going through (, 0) and (, ) in slope intercept form..) Find the equation of the line with x intercept and y intercept in standard form..) Find the equation of the line going through (, ) with x intercept 6 in standard form. B. Parallel and Perpendicular Lines For # 6, 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..) y x Slope of l : Slope of l : Parallel lines have slopes.
22 6.) y x Slope of l : Slope of l : 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: 7.) m = 8.) m = 7 9.) m = 0 Are the lines with these slopes parallel, perpendicular, or neither? 6, 0.).),.),.) Find the slope of a line parallel and perpendicular to AB where A(, ) and B (, )
23 For # 6, state whether the given pair of lines is parallel, perpendicular, or neither:.) y = x + 8x 6y =.) y = x + x + y = 9 6.) y = x + x = y + 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! ** 7.) Find the equation of the line going through (, ) and parallel to y = x + in slope intercept form. 8.) Find the equation of the line going through (, ) and perpendicular to x y = in standard form. 9.) Find the equation of the line going through (, ) and perpendicular to y = x + in slopeintercept form. 0.) Find the equation of the line going through (9, ) and parallel to x y = in standard form.
24 Notes #0: Review You can now solve linear systems (a set of 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.) y = x x + y = st Method: substitution or elimination nd Method: graphing (graph both lines on the coordinate plane below) y x solution: (, ).) x y = x + y = 6 st Method: substitution or elimination solution: (, ) nd Method: graphing (graph both lines on the coordinate plane below) y x
25 Chapter Study Guide For #, identify whether the angles are vertical angles, same side interior angles, corresponding angles, alternate interior angles, same side exterior angles, or alternate exterior angles..) and 6.) and 6 7.) and.) and 7 6 For # 6, find the slope of the line passing through the two points.) (, ) and (, ) 6.) ( 9, ) and (, ) 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) b) / For #8 0, name the two lines and transversal that form each pair of angles: 8.), lines:, trans: B C 9.) BAD, CDA lines:, trans: 0.) BAD, lines:, trans: A D E In the diagrams, the lines shown are parallel. Write an equation and solve for x and y. (The answer to # is two fractions.) Justify your work..).) y+0 x+80 x+0 x+0 y+0 y+6 Find the values of x and y..).) 0 y y x 0 0 x x
26 6 Define your variables, write an equation, and solve:.) The sum of two numbers is 0. The difference of the first number and twice the second number is. 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. 6.) 0 7.) 7 0 A B 8.) 9.) 9 0 F E C 6 D Solve for x and y: 0.) Solve for x and y:.) x + y 0 0 6x y x + y = x y = 8.) 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: Prove: m n Statements.).) Reasons m n.).).).).).)
27 .) Given: l m 7 Prove: m + m = 80 l m.) Statements.) Reasons.).).).).).).) Given: AB = CD AE = FD A E B Prove: EB = CF C F D.) Statements.) Reasons.).).).).) AE = FD.).).) For # 6, classify the triangles based on their sides and their angles:.) 6.).).) 8x 0 x + x 60 60
28 8 Use the diagram for reference. Show all equations and work. 7.) If m 6 = and m 8 = 6, then m 0 = ) If m 6 = 7 x, m 7 = x +, and m = 6 x +, then 9.) x =. If m 8 = 7x, m 7 = x 7, and m 9 = 0 x +, then x =. For #0, a, b, c, and d are distinct coplanar lines. How are a and d related? 0.) a b, b c, c d.) a b, b c, c d For #, find the measure of an interior angle and an exterior angle of each regular polygon..) an octagon.) a pentagon Complete the table for regular polygons.) (a) (b) (c) Number of Sides 6 Sum of exterior angles Measure of each exterior angle 0 Measure of each interior angle 6 Sum of interior angles Work: (a) (b) (c)
29 Graphing Linear Equations:.) Graph each line using the slope and y intercept: a) y = x + b) x + y = c) x y = 8 y x Find the x and y intercept of each line: a) x + y = 6 b) x y = 8 c) y = x + 6.) Graph the lines using the x and y intercepts: a) x + y = 6 b) x y = c) x y = 0 y x Find the slope and y intercept of each line: a) y = x b) y = c) x y = 9 9. Find the intersection of the two lines using the substitution or elimination method. x + y = 8 and x + y = 0 0. Explain why these two lines will not intersect y = x and 8x y = 6 Writing Linear Equations: Write an equation of the line with:. y intercept and slope in standard form. x intercept and y intercept in slopeintercept form. through (, ) with slope in slopeintercept form. through (6, ) and parallel to x y = in standard form
30 0. through (, ) and perpendicular to x + y = 7 in standard form 6. through (, ) and (, 7) in slope intercept form 7. through (, ) and (7, ) in slope intercept form 8. through (, ) and with x intercept in standard form 9. x intercept and y intercept in standard form 0. through (, ) and perpendicular to x y = in slope intercept form For #, are the given lines parallel, perpendicular, or neither?.) y = x + x 6y = 0.) y = x + x + 6y = 0.) y = x + 6x y = 0
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