Understanding Parallel Lines With Solved Examples
Example 1: In the following figure, find all other angles if 5 = 70 degrees.
Solution: Given 5 = 70°
So, 5 = 1 = 70°
1 = 4 = 70°
5 = 7 = 70°
Now, 1+ 2 = 180°
So, 70° + 2 = 180°
2 = 180° 70°
2 = 110°
Now that we have found the value of 2, we find the remaining angles as follows
2 = 6 =110°
2 = 3 = 110°
6 = 8 = 110°
Example 2: If AB is parallel to CD, find the value of x in the following figure if the value of y = 130°.
Solution: Given that AB is parallel to CD and y =130°
So, 4x+10 = y = 130°
4x+10 = 130°
x = 30°
Parallel Lines And Transversal
When any two parallel lines are intersected by another line called a transversal, many pairs of angles are formed. While some angles are congruent , the others are supplementary. Observe the following figure to see the parallel lines labeled as L1 and L2 that are cut by a transversal. Eight separate angles have been formed by the two parallel lines and a transversal. Each angle has been labeled using an alphabet.
Given below are the pairs of angles formed by the two parallel lines L1 and L2.
- Corresponding Angles: It should be noted that the pair of corresponding angles are equal in measure. In the given figure, there are four pairs of corresponding angles, that is, a = e, b = f, c = g, and d = h
- Alternate Interior Angles: Alternate interior angles are formed on the inside of two parallel lines that are intersected by a transversal. They are equal in measure. In this figure, c = e, d = f
- Alternate Exterior Angles: Alternate exterior angles are formed on either side of the transversal and they are equal in measure. In this figure, a = g, b = h
- Consecutive Interior Angles: Consecutive interior angles or co-interior angles are formed on the inside of the transversal and they are supplementary. Here, c + f = 180°, and d + e = 180°
- Vertically Opposite Angles: Vertically opposite angles are formed when two straight lines intersect each other and they are equal in measure. Here, a = c, b = d, e = g, f = h
Angles And Parallel Lines Theorems
We can make statements regarding parallel lines based on the angles they form. In other words, we can prove that lines are parallel based on angles, and conversely, we can also prove angle congruency based on the existence of parallel lines. Before proceeding further, let’s review some basic definitions and concepts regarding parallel lines. First, how can we tell the difference between parallel lines and those that are non-parallel?
Non-parallel lines are two or more lines that are not at equal distance and which are intersecting at some point or which will intersect at some point.
Non-parallel lines, StudySmarter Originals
You may be wondering, how do parallel lines relate to angles if they never intersect? The answer is transversals: Transversal lines play an important role in determining the angles associated with parallel lines.
A line passing through two lines at different points in the same plane is called a transversal line.
Transversal, Studysmarter Originals
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Parallel Lines: Definition Properties Examples
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In geometry, a line is a one-dimensional figure that has no initial point and no endpoint. Therefore, we represent a line with arrows on both ends to denote that it continues forever for both sides. But what are parallel lines? As the name suggests, these lines run parallel to each other. Parallel lines do not meet or intersect at any point. But why do we study parallel lines geometry, and what are their applications? Read on to learn more about line segments in detail below.
Heres what well cover:
- Parallel Lines Definition
- Parallel Lines Cut by a Transversal
- Properties of Parallel Lines
- Applications of Parallel Lines in Real Life
Parallel lines definition: Two lines are said to be parallel if they do not intersect each other at any point in the plane. Parallel lines are equidistant from each other throughout.
Furthermore, in 3-D Euclidean space, parallel lines also maintain a constant separation between points closest to each other on the two lines in addition to never intersecting one another.
Symbol of parallel lines: The symbol representing parallel lines is ||.
Note: Parallel lines are not always equal in length, but they are always at an equal distance from each other.
Parallel lines Examples from Real Life
If you observe your surroundings, you will encounter several parallel lines. Some common examples that you must watch next time are:
Angles Formed By A Transversal
In this figure, the pairs of angles formed by a transversal and parallel lines are shown. They are:
- alternate interior angles
- alternate exterior angles
- corresponding angles
Those pairs are listed in the table below.
|Alternate interior angles: 46, 35||Angles between || lines and on opposite sides of the transversal|
|Alternate exterior angles: 17, 28||Angles on the outside of || lines and on opposite sides of the transversal|
|Corresponding angles: 15, 48, 26, 37||Angles in corresponding positions of their respective intersections, e.g. upper-left to upper left|
|Interior angles on the same side: 45, 36||Angles between || lines and on the same side of the transversal|
|Exterior angles on the same side: 18, 27||Angles on the outside of || lines and on the same side of the transversal|
|Vertical angles: 13, 24, 57, 68||Angles opposite one-another at intersections, e.g. 13|
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What Letters Have Parallel And Perpendicular Lines
There are some letters in the English alphabet that have parallel and perpendicular lines in them. Observe the horizontal lines in E and Z and the vertical lines in H, M and N to notice the parallel lines. Similarly, observe the intersecting lines in the letters L and T that have perpendicular lines in them.
Angles Parallel Lines And Transversals
Two lines that are stretched into infinity and still never intersect are called coplanar lines and are said to be parallel lines. The symbol for “parallel to” is //.
If we have two lines and have a third line that crosses them as in the figure below – the crossing line is called a transversal:
In the following figure:
If we draw to parallel lines and then draw a line transversal through them we will get eight different angles.
The eight angles will together form four pairs of corresponding angles. Angles F and B in the figure above constitutes one of the pairs. Corresponding angles are congruent if the two lines are parallel. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs.
Angles that are in the area between the parallel lines like angle H and C above are called interior angles whereas the angles that are on the outside of the two parallel lines like D and G are called exterior angles.
Angles that are on the opposite sides of the transversal are called alternate angles e.g. H and B.
Angles that share the same vertex and have a common ray, like angles G and F or C and B in the figure above are called adjacent angles. As in this case where the adjacent angles are formed by two lines intersecting we will get two pairs of adjacent angles that are both supplementary.
Two angles that are opposite each other as D and B in the figure above are called vertical angles. Vertical angles are always congruent.
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Parallel Lines And Slopes
Parallel lines are lines that do not intersect. In two dimensions, parallel lines have the same .
We can write the equation of a line parallel to a given line if we know a point on the line and an equation of the given line.
Write the equation of a line that passes through the point ( and is parallel to the line
Parallel lines have the same slope.
The slope of the line with equation y . So, any line parallel to y
Now use the to find the equation.
We have to find the equation of the line which has slope 2 and passes through the point (
is parallel to the line y and passes through the point (
How To Construct Parallel Lines
Construct parallel lines with a straightedge, a pencil, and plain paper. If using a ruler, this is a one-step process. Lay the ruler on the paper, holding it carefully, so it does not slip. Trace each edge of the ruler. You’re done!
When you move the ruler, you have parallel lines. You can label them, identifying each line with two named points and placing arrowheads at the end of the two lines.
You can also construct a line parallel to an existing line passing through a single point, not on the line. You need a drawing compass plus the plain paper, pencil, and straightedge.
This new line is far to the right of the first line but is exactly parallel because the x-value determined its slope. Same x-value same slope both lines parallel!
Slope Of Perpendicular Lines
When plotting perpendicular lines on a coordinate graph, you need to consider two ideas:
- The slopes will be opposites
- The slopes will be reciprocals
Let’s take the first requirement: opposite slopes. We’ll keep one of our earlier lines with a positive slope of 2, and then show a new, second line with a negative slope of -2:
Now the lines are crossing, with our new line showing x-values increasing as y-values are decreasing. Negative slopes have that inverse relationship between the x-values and y-values. But our intersecting lines are not perpendicular, yet.
The slopes must be reciprocal, so instead of simply having one with a positive slope of 2 and one with a negative slope of -2, we need the second line to be – 1 ):
Like parallel lines, examples of perpendicular lines surround us, in walls meeting floors and ceilings, in floor tiles, in bricks in walls, in window grilles. The margin line on a sheet of notebook paper is perpendicular to the parallel writing lines.
Transverse Lines Or Transversals
Coplanar lines that intersect other coplanar lines are called transverse lines or transversals. A transversal crossing two parallel lines creates eight angles, which can be viewed and compared many ways:
Transverse lines are everywhere in nature and human-made objects. Street maps show parallel, perpendicular and transverse lines. Mown hay lies in bundles of transverse lines, with any three strands being coplanar. A handful of casually tossed pencils will criss-cross as transverse lines.
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What Letter Has Both Parallel And Perpendicular Lines
There are some letters in the English alphabet that have both parallel and perpendicular lines. For example, the letter H, in which the vertical lines are parallel and the horizontal line is perpendicular to both the vertical lines. Similarly, in the letter E, the horizontal lines are parallel, while the single vertical line is perpendicular to all the three horizontal lines.
Properties Of Parallel Line
When a transversal line cuts two lines, the properties below will help us determine whether the lines are parallel.
1. Two lines cut by a transversal line are parallel when the corresponding angles are equal.
The two pairs of angles shown above are examples of corresponding angles. In general, they are angles that are in relative positions and lying along the same side.
2. Two lines cut by a transversal line are parallel when the alternate interior angles are equal.
Alternate interior angles are a pair of angles found in the inner side but are lying opposite each other.
3. Two lines cut by a transversal line are parallel when the alternate exterior angles are equal.
Alternate exterior angles are a pair of angles found on the outer side but are lying opposite each other.
4. Two lines cut by a transversal line are parallel when the sum of the consecutive interior angles is $\boldsymbol}$.
Consecutive interior angles are consecutive angles sharing the same inner side along the line.
5. Two lines cut by a transversal line are parallel when the sum of the consecutive exterior angles is $\boldsymbol}$.
Consecutive exterior angles are consecutive angles sharing the same outer side along the line.
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Parallel Lines Cut By A Transversal
In geometry, a transversal is a line that intersects two or more lines. When a transversal cuts two parallel lines, several angles are formed. The angles are related as follows:
Exterior angles The angles that are situated outside of the line. Here, 1, 2, 7, and 8 are exterior angles.
Interior angles The angles located inside of the lines are. Here 3, 4, 6, and 5 are interior angles.
Corresponding angles The angles that occupy the same relative position at each intersection. Here,
- 1 and 5
- 4 and 8
- 3 and 7 are corresponding pairs of angles.
Alternate Interior angles- These angles are formed on the inside of two parallel lines when a transversal intersects them. The pairs of alternate interior angles are
- 3 and 5
- 4 and 6
Exterior Alternate angles These angles are formed on either side of the transversal outside the two parallel. The pairs of alternate exterior angles are
- 1 and 7
- 2 and 8
Co-interior angles These are also called interior angles on the same side or consecutive interior angles, or allied angles. Here, the co-interior angles are
- 4 and 5
Vertically opposite angles: The pairs of vertically opposite angles are
- 1, and 3
The basic properties of parallel lines are as follows:
- They do not intersect with each other.
- Parallel lines do not meet, or they meet at infinity.
- They are always equidistant from each other.
How to Test If Two Lines are Parallel?
What Are Parallel Lines
In geometry, parallel lines can be defined as two lines in the same plane that are at equal distance from each other and never meet.
Here, three set of parallel lines have been shown vertical, diagonal and horizontal parallel lines.
Sides of various shapes are parallel to each other. Parallel lines are represented with a pair of vertical lines between the names of the lines, such as PQ XY.
We can see parallel lines in a zebra crossing, the lines of notebook and in railway tracks around us.
|Fun Facts1. Each line can have many parallel lines to it. 2. Parallel lines can be extended indefinitely, with out them intersecting at any point.|
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What Are Parallel Lines And Perpendicular Lines
Parallel lines are those lines that are equidistant from each other and never meet, no matter how much they may be extended in either directions. For example, the opposite sides of a rectangle represent parallel lines. On the other hand, if any two lines intersect each other at 90°, they are called perpendicular lines. For example, the adjacent sides of a rectangle are perpendicular lines because they intersect each other at 90°.
Alternate Exterior Angles In A Transversal
A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal.
In the above figure, 2 and 8 form a pair of alternate exterior angles. Another pair of alternate exterior angles in this figure is 1 and 7.
Now, we shall learn about the alternate interior angle theorem that is also one of the basic properties of alternate interior angles.
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Spherical Or Elliptic Geometry
In spherical geometry, all geodesics are great circles. Great circles divide the sphere in two equal hemispheres and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of the sphere.
If l, m, n are three distinct lines, then l .
In this case, parallelism is a transitive relation. However, in case l = n, the superimposed lines are not considered parallel in Euclidean geometry. The binary relation between parallel lines is evidently a symmetric relation. According to Euclid’s tenets, parallelism is not a reflexive relation and thus fails to be an equivalence relation. Nevertheless, in affine geometry a pencil of parallel lines is taken as an equivalence class in the set of lines where parallelism is an equivalence relation.