In the previous graph, the slope triangles △ABF and △CDE can be mapped onto each other by a translation. Since translations are rigid motions, it can be concluded that △ABF and △CDE are congruent triangles. Because the slope triangles are congruent, the slopes of the lines are equal. This means that the lines are parallel.
Let and be parallel lines, and m1 and m2 be their respective slopes. Then, the following statement is true.
The slope of a vertical line is not defined. Therefore, this theorem only applies to non-vertical lines. Any two distinct vertical lines are parallel.
Since the theorem consists of a biconditional statement, the proof will consists in two parts.
Both directions of the biconditional statement have been proved.
The first transatlantic telegraph cable was laid between Valentia in western Ireland and Trinity Bay Newfoundland in the 1850s. With the invention of fiber optic cables, the number of transatlantic cables has increased. The following map shows some of these cables.
By observing the graph, it can be seen that the line passes through the points (0,3) and (4,4). When a slope triangle is constructed between these points, the slope of the line is calculated as
Recall that a system of linear equations can have infinitely many solutions, one solution or no solution. Under which conditions does a system of linear equations have infinitely many solutions or no solution?Consider the following systems of equations.
System I: Infinitely many solutions
System II: No solution
From the previous example, the following conclusions about systems of linear equations can be made.
|The lines of the system have the same slope and the same y-intercept.||The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.|| |
|The lines of the system have the same slope but different y-intercept.||The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.|| |
|The lines of the system have different slopes.||The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.|| |
These conclusions can be seen in the following diagram.
If and are two perpendicular lines with slopes m1 and m2, respectively, the following relation holds true.
Since the theorem is a biconditional statement, it will be proven in two parts.
The biconditional statement has been proven.
Just like with parallel lines, there is an infinite number of perpendicular lines to a given line. Two pieces of information will help in writing the equation of a perpendicular line. These are the slope — which is calculated using the slope of the given line — and a point that lies on the line.The equation of a line is given in standard form.
It has been discussed how to find the equation of a line that is perpendicular to a line whose equation is given. What about finding the equation of a line perpendicular to a line whose graph is given?Mark and Paulina have been asked to write an equation of a line perpendicular to the line shown in the graph. Additionally, they are told that this perpendicular line should pass through the point
The theorems seen in this lesson can be used to identify quadrilaterals and some of their properties.
Determine whether quadrilateral ABCD is a parallelogram. Explain the reasoning.
Yes, see solution.
Yes, see solution.
Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.