∠5 because an angle supplementary to a second angle must be supplementary to any other angle of the same measure.Ħ. Using the numbered diagram above, we see that ∠1 ≅∠5 by the corresponding angle postulate, and ∠4 is supplementary to ∠1 (obvious from the diagram). Interior angles on the same side are supplementary Therefore alternate exterior angles are congruent.ĥ. (Figure ↑) $\angle 1 \cong \angle 5$ by the corresponding angle postulate, and $\angle 5 \cong \angle 7$ because vertical angles are congruent, therefore $\angle 1 \cong \angle 7$ by substitution of $\angle 7$ for $\angle 5$ in the first expression.
Therefore alternate interior angles are congruent. We will continue our discussion the next post.(Figure ↑) $\angle 1 \cong \angle 5$ by the corresponding angle postulate, and $\angle 1 \cong \angle 3$ because vertical angles are congruent, therefore $\angle 3 \cong \angle 5$ by substitution of $\angle 3$ for $\angle 1$ in the first expression. Can you see the relationship between the two equations? So, if two equations can be transformed to one form and they are equal, they are coinciding lines.īut how about 3x + 8y = 12 and 6x + 16y = 24? They are coinciding lines. For example, the lines 3x + y = 5 and 3x = 5 – y are coinciding lines because they both can be transformed to y = -3x + 5. Therefore, to be able to distinguish coinciding lines using equations, you have to transform their equation to the same form (e.g. Therefore, if two lines on the same plane have different slopes, they are intersecting lines.Įxercise: Give equations of lines that intersect the following lines.Īs discussed above, lines with the same equation are practically the same line. The extension of the line segments are represented by the dashed lines. If you extend the two segments on one side, they will definitely meet at some point as shown below.
You can imagine this by drawing two line segments that are not parallel on a board (the intersection may be outside the board). If two lines have different slopes, then definitely they will meet at a certain point. Therefore, for distinct lines to be parallel, their slope (m) must be equal and their y-intercept (b) must be different.Įxercise: Give equations of lines that are parallel to the following lines. If we have two lines having the same slope and intersect the same point on the y-axis, then they are practically the same line (Can you visualize why?). īut what if they have the same b? Are they still parallel? For example, the lines with equationsĪre parallel lines because both of them have. Therefore, we will know that two lines are parallel if they have the same value for when the equations are of the form. The y-intercept is the point in the y-axis where the line intersects. In linear equations, recall that in the slope-intercept form, is the slope of the line and is the y-intercept. What we know about parallel lines is that they have same slope. Can we determine if lines are parallel, intersecting, or coinciding based on equations only? Parallel Lines
We can examine the three cases mentioned above in terms of equations. We have learned that linear equations can be represented by, where and are real numbers. The equations which represent lines are called linear equations. In Algebra, we have learned that a line can be represented with an equation.
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