ACT study guide Math: Parallel and Perpendicular Lines
Which of the following is the equation of a line parallel to the line passing through the points (−1, 4) and (3, 8)?
(The slope of the line passing through the given points is 8−4/3−(−1)=4/4=1. Any line parallel to this line will also have a slope of 1. The answer choices are of the form y = mx + b, so the coefficient of x will be the slope. The only equation in which the coefficient of x is 1) y = x − 10
If lines AB and CD are perpendicular and intersect at point P, what is the measure of ∠APD in degrees?
(This problem does not provide a figure, because you don't need one to answer the question. By definition, the angle between two perpendicular lines is )90°
(Since the angle between the lines is 90°, the lines are perpendicular, and the slopes are negative reciprocals of each other.) -1
As shown in the following figure, the angle between lines m and ℓ is 90°. What must the product of their slopes equal?
(Since all the angles in the larger triangle must add up to 180º degrees, ∠ ADE = 180º − 30º − 65º = 85º. Line AD may be considered as a transversal through the two parallel lines; therefore, ∠ ADE is congruent to ∠ AxC = 85º. Angle x is supplementary to AxC because the two angles form a line. Therefore,) x = 180º − 85º = 95º
BC is parallel to DE in the figure. If ∠DAE is 65° and ∠AED is 30°, what is the value of angle x? Note: Figure is not to scale
In order for a system of 2 linear equations to have no solutions, the graphs of the equations must be parallel. Parallel lines have the same slope. To find the equation whose graph is parallel to the line in the figure, you must find the slope of the line between the points (0, −4) and (3, −0). Since slope is rise/run, the slope is 4/3. The only equation with the correct slope of 4/3 is y=4/3x+2.
If a system of 2 linear equations in 2 variables has NO solution, and 1 of the equations is graphed in the (x, y) coordinate plane below, which of the following could be the other equation?
What is the slope of a line that is parallel to the line determined by the equation 5x − 4y = 8?
Parallel lines have equal slopes. To find the slope of a line that is parallel to the line determined by the equation 5x − 4y = 8, put the equation in slope-intercept form (y = mx + b). To do so, first subtract 5x from both sides to get −4y = −5x + 8. Then divide by −4 to get y=5x/4−2. Thus the slope is 5/4.
(A line through AB would have a slope 9−0/0−3=−3, and any line parallel to this line will have the same slope by definition.) -3
Which of the following is the slope of a line that is parallel to the line that would pass through the points A and B in the figure shown?
Which of the following statements must be true of a line in the (x, y) coordinate plane defined by the equation x = c for some constant c? I. The line is parallel to the y-axis. II. The line is perpendicular to the x-axis. III. The line has an x- and y-intercept of c.
(Any line of the form x = c is parallel to the y-axis, which is equivalent to being perpendicular to the x-axis. The x-intercept is c, but there is no y-intercept because the line is parallel to the y-axis, so III is not true.) I and II only
Which of the following equations represents a line that is parallel to the x-axis?
(If a line is parallel to the x-axis, it will be of the form y = c for some constant c.) y = 18
If the x- and y-intercepts of a line in the (x, y) coordinate plane are nonzero and share the same sign, which of the following statements MUST be true?
(If both are positive, then the line will have to fall from left to right. If both are negative, the same is true.) The slope of the line is negative
What is the slope of a line that is perpendicular to the line determined by the equation 7x + 4y = 11?
(Perpendicular lines have slopes that are opposite reciprocals. To find the slope of a line perpendicular to 7x + 4y = 11, first find the slope by converting the equation to slope-intercept form, then take the opposite reciprocal. To do so, first subtract from both sides to get 4y = −7x + 11. Next, divide both sides by 4 to get y=−7x/4+11/4. Since the slope in this line is −7/4, the slope of a line perpendicular to that is )4/7
In the standard (x, y) coordinate plane, which of the following lines goes through (3, 4) and is parallel to y = 2x + 2?
(Remember that all parallel lines have the same slope, so a line parallel to y = 2x + 2 will have a slope of 2. A quick way to aid you in solving this problem would be to eliminate answer choices that do not have slope 2, so answer choices A and E can be immediately eliminated. Check the point (3, 4) in the remaining answer choices. The only choice that works is )y = 2x − 2
Three distinct lines, all contained within a plane, separate that plane into distinct regions. What are all of the possible numbers of distinct regions of the plane that could be separated by any such three lines?
(Start by drawing 3 parallel lines. Now, try drawing 3 lines in other configurations, and you will see that there will always be either 6 or 7 regions Therefore, the correct answer is) 4, 6, or 7
Which of the following equations represents a line that is perpendicular to the line 5x − 2y = 8?
(The slope of a line in the form ax + by = c is −(a/b). In this case, the slope is 5/2, and any line perpendicular to this line will have a slope of −2/5. Only the equation in choice E satisfies this condition.) −2x − 5y = 10
What is the slope of a line that is perpendicular to 8x + 4y = 2 ?
(The slope of the given line can be found by solving for y. The result is y=1/2−2x, which has a slope of −2. The slope of any perpendicular line will be the negative reciprocal of −2, which is) 1/2.
A system of linear equations is shown below. 4y− 2x= 8 4y+ 2x= 8 Which of the following describes the graph of this system of linear equations in the standard (x, y) coordinate plane?
(To solve this problem, it would be helpful to make the properties of both lines more evident by converting them to slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept). To start, adding 2x to both sides and dividing both sides of the equation 4y − 2x = 8 by 4 yields y=x/2+2 . Similarly, subtracting 2x from both sides and dividing both sides of the equation 4y +2x =8 by 4 yields y=−x/2+2 . The relationship between y=x/2+2 and y=−x/2+2 is that they are lines that share the same y-intercept, yet they have opposite slopes (1/2and−1/2). Therefore these are two distinct intersecting lines.) Two distinct intersecting lines
What is the slope of any line parallel to the line 2x−3y=7?
(parallel lines always have the same slope. Remember that to find the slope of the line, you should convert the equation 2x −3y = 7 into slope-intercept form y = mx + b, where m is the slope: 2x − 3y = 7 Subtract 2x. −3y = 7 − 2x Divide by −3. y = 7−2x−3 Separate the fraction. y = 7−3−2x−3 Reorder and reevaluate negatives. y = 23x−73 Find the m term. Thus the slope of this line, or any line parallel to it, is) 23
(Because lines m and n are parallel, angles x° and y° must have the same value; that is, x° = y°. So x° + y° = x° + x° = 2x° = 160°, and x° = 80°. The angles x° and z° must have a sum of 180°, so z = )100°
In the following figure, lines m and n are parallel, and line ℓ is a transversal crossing both lines. If the sum of x° and y° is 160°, what is the value of z?
(Since m and n are parallel, the sum of angles x and y must be 180°, and since x is 36°, y = 180° − 36° =) 144°
In the following figure, lines m and n are parallel, and the value of x is 36°. What is the value of y?
(Focus on the small triangle at the bottom of the diagram. The angle on the bottom left of the triangle is congruent to x (alternate interior). The angle on the bottom right is congruent to y (vertical angles). Since all angles in the triangle add up to 180°, the top angle in the triangle must measure 180° − x − y. Angle z is supplementary to that top angle, so z = 180° − (180° − x − y) = x + y) z = x + y
Line A is parallel with line B as shown in the diagram. Given the angles x and y, which of the following is a valid expression for angle z in terms of angles x and y?
(Remember, do not assume that the figures are drawn to scale. Since the lines m and n are parallel, y = 75° (corresponding angles). And since the angles lie on the same line, y + 3x = 180°. Therefore,) 75° + 3x = 180°, and x = 35°
Line ℓ is a transversal of the parallel lines m and n. The measures of two angles are given in terms of x and y. What is the value of x, in degrees?
(The line parallel with line AB must have the same slope as line AB The line rises 2 units and goes over 3 units from point A to point B, so the slope is 2/3 . So answers A and C are the only options. Point C at (6, 2) satisfies) y = 2/3x − 2
The graph shows a line going through points A and B. Which of the following equations represents the line passing through point C that is parallel with line AB?
(The slope of the line given is −3/2, so the slope of a line perpendicular to that is + 2/3. Going up 2 units and over 3 units from (−2, −1) gives [(−2 + 3), (−1 + 2)] = ) 1, 1
The points (−2, −1) and (x, y) define a line that is perpendicular to the line shown in the diagram. Select a possible point (x, y)
