Unit 3 Notes: Parallel Lines, Perpendicular Lines, and Angles 3-1 Transversal REVIEW: *Postulates are Fundamentals of Geometry (Basic Rules) To mark line segments as congruent draw the same amount of tic marks on each one. Naming the angle in 4 ways: A 1) B 3) 1 2) ABC 4) CBA 1 * Transversal a line that intersects two unique lines at two different points B C There are five types of angles with regard to a transversal *Corresponding angles: Pairs of angles that are in the same location with 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 transversal between the intersected lines. 3 and 6, 4 and 5 *Alternate Exterior Angles: Pairs of angles on opposite sides of the transversal outside the intersected lines. 1 and 8, 2 and 7 *Consecutive Interior Angles: Pairs of angles on the same side of the transversal Between 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 transversal outside the intersected lines. aka same side exterior 1 and 7, 2 and 8, 1
Using Transversal a Identify all pairs of Alternate Interior Angles: 4& 5, 3 & 6 Alternate Exterior Angles: 1 & 8, 2 & 7 Consecutive Interior Angles: 3 & 5, 4& 6 Consecutive Exterior Angles: 1 and 7, 2 and 8 Corresponding Angles: 1& 5, 2& 6, 3& 7, 4& 8 Using Transversal b Identify all pairs of Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Consecutive Exterior Angles Corresponding Angles Using Transversal c Identify all pairs of Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Consecutive Exterior Angles Corresponding Angles 2
3-2 Properties of Perpendicular Lines Perpendicular 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. g g h h Perpendicular Complementary Theorem: If two sides of two adjacent acute angles are perpendicular, Then the angles are complementary. Developmental Proof of Linear Perpendicular Theorem: Statement Reason 1) 1 2 1) Given 2) 2) Given: 1 2, 1 & 2 are a linear pair. 3) 1 & 2 are a linear pair 3) Given 4) 4) Linear Pair Post Prove: g h 5) m 1 + m 1 = 180 5) Sub. Prop. of = g 6) 6) 7) m 1 = 90 7) Div. Prop of = 8) 8) 9) g h 9) Def. of Lines 1 2 h 3
3-3 Properties of Parallel Lines FOLLOWING THEOREMS ARE USEFUL IN PROOFS DEALING WITH PARALLEL LINES ***Corresponding Angles Postulate: - If a transversal intersects two parallel lines, then the corresponding angles formed are congruent. If a b, then all corresponding angles are congruent 1 5, 2 6, 3 7, 4 8 If 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 8= m 3= m 5= m 7= *Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then the Alternate Interior Angles are congruent. If a b, then all pairs of Alternate Interior Angles are congruent. 3 6, 4 5 If 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 the Alternate Exterior Angles are congruent. If a b, then all pairs of alternate exterior angles are congruent. 1 8, 2 7 If m 1 = 32, then what is m 8? 4
*Consecutive Interior Angles Theorem If a transversal intersects two parallel lines, then the 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, then the 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 Postulate Statements 1) a b 1) Reasons 2) 4 1 2) 3) 1 5 3) 4) 4 5 4) 5
Examples using geometric shapes: Find the value of x. 3-4 Proving Lines Parallel Converse theorems about transversals and parallel lines ***Converse to the corresponding angles postulate If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel *** If 1 5, then a b If m 1 = 2x + 66 and m 5 = 8x -24, then what is the value of x? and what is the measure of each angle? ***Converse to the Alternate interior angles theorem -If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.*** If 2 5, then a b If m 2 = 3x -4 and m 5 = 2x + 16, then what is the value of x?... and what is the measure of each angle? 6
***Converse to the Alternate Exterior Angles Theorem-If two lines and a transversal form alternate exterior angles that are, then the lines are *** If 3 7, then a b If 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 form consecutive 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 of each angle? ***Converse to the Consecutive Exterior Angles Theorem -If two lines and a transversal form consecutive exterior angles that are, then the lines are.*** If m 1 + m 7 =, then 1 a 7 b If m 1 = 6x - 14 and m 7 = 3x 13, then what is the value of x? and what is the measure of each angle? 7
3-5 Transversal Parallel and Perpendicular Line Theorems Transitive 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 each other.*** If a d and b d, then a b Developmental proof of the parking lot theorem. Statements Reasons 1) a d, b d 1) 2) 1 and 2 are right s 2) 3) 1 2 3) 4) a b 4) Statements Reasons 1) a b, b c, and c d 1) 2) a c 2) 3) a d 3) 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 the other.*** If a b and a d, then b d A 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. (DRAW A PICTURE) 1) a b, b c, c d What 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 Proof Given: m 1 m 2 Prove: line k is not perpendicular to line m. 1 2 k m 9
3-6 Lines in the Coordinate Plane Finding an Equation of a Line The Point-Slope Equation A non-vertical line with slope m and containing a point(x 1, y 1 ) has the point-slope equation y y 1 = m(x x 1 ) Example 1 Write an equation for the line with slope 3 that contains the point (5, 2). Express the equation in slope-intercept form. y y 1 = m(x x 1 ) y 2 = 3(x 5) y 2 = 3x 15 y = 3x 13 Try This. Write an equation for each line with the given point and slope. Express the equation in slope-intercept form. a. (3, 5), m = 6 b. (1, 4), m = 2 3 We can also use the point-slope equation to find an equation of a line if we know any two points on the line. Example 2. We first find any two points on the line. Use (1, 1) and (2, 3). We next find the slope. m = 3 1 2 1 = 2 1 = 2 10
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 y 1 = m(x x 1 ) y 1 = 2(x 1) y 1 = 2x 2 y = 2x 1 Try This. Write an equation for each line in slope-intercept form. g. h. 11
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. 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 + 4 6x + 2y = -10 2y = -6x 10 y = -3x 5 The 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 = 10 2) 4y = -12x + 16 and y = 3x + 4 Perpendicular Lines are lines that intersect to form a 90 angle (or a right angle). A vertical line and a horizontal line are perpendicular. Algebraically, the product of the slopes of perpendicular lines is -1. The slopes are 2 and 1 2 have a product of -1 so these lines are perpendicular. 12
Example 2 Tell 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. 3y = 9x + 3 6y + 2x = 6 The slopes are 3 and 1 3 Y= 3x + 1 6y = -2x + 6 Y = 1 3 x + 1 The products of the slopes of these lines is 3(- 1 ) = -1. Thus the lines are perpendicular. 3 Try This. Tell whether the graphs of the equations are perpendicular. 1) 2y x = 2 and y = -2x + 4 2) 4y = 3x + 12 and -3x + 4y 2 = 0 Example 3 Write an equation for the line containing (1, 2) and perpendicular to the line y = 3x 1. The slope of the line y = 3x 1. The slope of the line y = 3x 1 is 3. The negative reciprocal of 3 is - 1 3. y y 1 = m(x x 1 ) y 2 = 1 (x 1) 3 y 2 = 1 3 x + 1 3 y = 1 3 x + 7 3 13
Try This. Write an equation for the line containing the given point and perpendicular to the given line. a. (3, 2); y = 2x + 4 b. (-1, -3); x + 2y = 8 14