Unit 6 Lesson 4: Parallel and Perpendicular Lines Flashcards
coordinate plane
a two-dimensional plane formed by the intersection of two number lines: the x-axis and the y-axis
rise
the number of units you move up or down from point to point; also known as the change in y, or Δy
run
the number of units you move left or right from point to point; also known as the change in x, or Δx
x-coordinate
the coordinate that identifies the exact location of a point on or parallel to the x-axis
slope
the measure of the steepness of a line
y-coordinate
the coordinate that identifies the exact location of a point on or parallel to the y-axis
How to see if something has a postive slope
The line in the first graph has a positive slope because it goes up from left to right, which illustrates an uphill slope.
How to see if something has a negative slope
The line in the second graph has a negative slope because it goes down from left to right, which illustrates a downhill slope.
what has a slope of 0
The line in the third graph has no steepness, so it has a slope of 0.
What is an example of an undefinded slope
The line in the fourth graph is so steep that it cannot be measured, so it has an undefined slope.
how to find the rise and the run
To find the height (or rise) of the ramp you created, you must calculate the difference between the y-coordinates (change in y). To find the length (or run) of that ramp, you must calculate the difference between the x-coordinates (change in x).
What is the equation for slope
Mathematically, an expression for the change in y is y2−y1
. Similarly, an expression for the change in x is x2−x1
. The slope of a line is equal to the ratio riserun
, so the formula for finding the slope of any line is change in ychange in x=y2−y1x2−x1=ΔyΔx
What is the slope of a line that
the two x-coordinates are the same
What happens in a line thats has 2 of the same y-coordinates
a 0 slope because the two y-coordinates are the same.
parallel
extending in the same direction, everywhere equidistant and not intersecting
parallel lines
a pair of lines on the same plane that never intersect
constant
a fixed value in the form of a number or a letter such as a, b, or c
coefficient
a number in front of a variable
slope-intercept form
the equation y=mx+b
where m is the slope of the line and b
is the y
-intercept
proof by contradiction
a method of proving a statement by assuming the statement is false, writing a proof to show that it is false, and then running into a contradiction or impossibility which shows that the statement is in fact true
How to prove 2 lines are paralell using proof of contradtion
we want to prove that two distinct lines with the same slope are parallel. To do a proof by contradiction, we will assume that two distinct lines with the same slope are not parallel.
Let l
and n
be two lines with the same slope but are not parallel. Let l
have the equation m=mx+b
. In order for line n
to be parallel to line l
but also be distinct, it must have the same slope but a different y-intercept. Therefore the equation of line n
is y=mx+c
where b≠c
.
Since the lines are not parallel, they have a point of intersection. Find the x
-coordinate of this point by setting the equations equal to each other and solving for x
.
mx+bmx−mx+bb===mx+cmx−mx+cc
This contradicts the condition that b≠c
. Therefore there is no point of intersection, so the lines are parallel.
We have proven that the statement that two distinct lines with the same slope are not parallel is false. Likewise, it has to be true that two distinct lines with the same slope are parallel. This concludes the proof.
Angle Addition Postulate
a hypothesis that states that if point D
lies in the interior of ∠ABC
, then m∠ABD+m∠DBC=m∠ABC
If two angles share a common vertex and ray, then the Angle Addition Postulate states that the sum of the measures of the two smaller angles is equal to the measure of the large angle formed by their outer rays.
Angle-Angle Similarity Postulate
a hypothesis that states that two triangles are similar if two of their corresponding angles are congruent
similar triangles –
triangles that have corresponding sides in the same ratio and have congruent corresponding angles
Similar Triangles
Similar triangles are triangles that have the same shape but different sizes. Their corresponding angles are congruent, and their corresponding sides are proportional. For example, if â–³ABC
and â–³EFG
are similar triangles, then the following is true:
∠A≅∠E
∠B≅∠F
∠C≅∠G
ABEF=BCFG=CAGE
Angle-Angle Similarity Postulate
If two triangles have two pairs of congruent angles, then the Angle-Angle Similarity Postulate states that the two triangles are similar.
