Intersecting Lines

When two lines cross each other they are known as Intersecting Lines.
These intersecting lines meet at a common point known as a Point of Intersection.

Observe the below image:
⟹ M & N are intersecting line
⟹ O is the point of intersection.

What are intersecting lines in Geometry

Multiple Line Intersection

There can be multiple lines which can intersect at the same point.

(i) Three Lines Intersection
Below image is of three lines intersecting at common point O

Intersecting Lines Examples

(ii) Four Line Intersection
Below image is of four lines intersecting at common point O

Learn about intersecting lines in Geometry for Math Class

Hence, there is no limit on number of intersection lines.
You can intersect as many lines as it seems possible to draw cleanly on paper.

Angle of Intersection

The angle at which the two lines intersect is called Angle of Intersection.

To find the angle of intersection, check the angle made by two lines at the point of intersection
The angle of intersection can be anywhere between 0 to 180 degrees.

Some examples are shown below:

Example 01

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In the above figure, line l1 and l2 intersect at point O.

Note there are two angles shown in the above image; 53 and 127 degree.
Both the angles are known as angle of intersection.

The difference is that:
⟹ 127 degree is the angle measurement from line l1 to l2
⟹ 53 degree is the angle measurement from line l2 to l1

Note: Always remember there are two angle of intersection. Just understand the location of the angles with respect to the lines drawn.

Example 02

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In the above image, line M & N are intersecting at point O at 90 degrees.

The two angles of intersection are:
90 degree ⟹ angle measured from M to N
90 degree ⟹ angle measured from N to M

Example 03

Intersecting lines definition

Given above are two lines M & N intersecting at point O

The angles of intersection are:
60 degree ⟹ angle measured from M to N
120 degree ⟹ angle measured from N to M

Angle Property of Intersecting Lines

(a) In two intersecting lines, the vertically opposite angles are equal

Intersection of two lines

In the above image, two lines intersect each other at point O.
Here the two angles \angle AON\ and\ \angle COB are vertically opposite and are equal in measure.

\angle AON\ = \angle COB\ = 50 degree

Similar is the case for other two angles

Angles of intersection

The other two angles \angle AOC\ and\ \angle BON are vertically opposite angles and are equal in measure.

\angle AOC\ = \angle BON\ = 130 degree

When two lines intersect, two sets of vertically opposite angles are formed which are equal in measure.

(b) When lines intersect at common point, the sum of all angle formed is exactly equal to 360 degree

Vertically opposite angles in line intersection

If you take the above discussed image and add all the angles you will get:
⟹ 50 + 130 + 50 + 130
⟹ 360 degree

Example 02

How do lines intersect

Observe the above image.

Using the concept of vertically opposite angles, we infer that:
\angle AOP\ = \angle BOQ\ = 54 degree
\angle AON\ = \angle COB\ = 66 degree
\angle NOQ\ = \angle POC\ = 60 degree

Point of intersection of two lines

Adding all the angles, we will get:
⟹ 54 + 60 + 66 + 54 + 60 +66
⟹ 360 degree

Intersection at more than one Points

Till now we have seen examples of line intersection at one point.

But if there are more than two lines, it is possible to have more than one intersection point.

Intersection of more than two lines

In the above image there are three lines L, M and N

Line M & N intersect at common point O

But line L do the intersection away from point O
⟹ Line L intersect N at point X
⟹ Line L intersect M at point Y

Hence, there are multiple intersection points in the above image.

Then what about the angles made? Will they show same property discussed above?

Yes, the same angle property can be applied here.

More than two points of intersection

Taking each point of intersection as a separate case and applying the angle property as discussed above

Intersection point O
Using the Property of Vertically Opposite Angles
\angle 1\ = \angle 2\
\angle 3\ = \angle 4\

Intersection point X
Again using vertically opposite angle concept
\angle 5\ = \angle 6\
\angle 7\ = \angle 8\

Intersection point Y
Again using vertically opposite angle concept
\angle 9\ = \angle 10\
\angle 11\ = \angle 12\

Hence, each point of intersection follow the angle property discussed above.

Questions on Intersecting Lines

(01) Find angle b in the below image

Intersecting Lines questions with solution
Read Solution

Line L & M are intersecting at point O

We know that Vertically Opposite Angles are equal
\angle a\ =\ \angle b
\angle a\ = 105 degree

(02) Find angle c in the below image

Problems on intersecting lines
Read Solution

We know that M is a straight line.
Hence angle on M will add up to 180 degree
\angle a\ + \angle c\ = 180
\angle c\ = 180 – 100
\angle c\ = 80

(03) In the below image, line M & N are parallel lines intersected by line L.
Observe the below image and find angle b

Intersecting Lines Examples

Read Solution

\angle a\ = 97 degree {Vertically Opposite Angle}

Since line M & N are parallel;
\angle a\ = \angle b\ {Corresponding angles}

Hence, \angle b\ = 97 degree

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