Table of contents  
Important Terms Related to Lines and Angles  
What is an Angle?  
Related Angles  
Pairs of Lines  
Checking for Parallel Lines 
Imagine you're in a room right now. Take a look around. What do you see?
There are various things all around you, like tables, chairs, a wall clock, a laptop, a water bottle, and even stairs if you have them.
When you peek outside, there are more objects like trees, roads, buildings, and electric poles.
Now, do you know that all these things are actually made up of simple shapes, like lines and angles?
Let's explore the world of shapes and how they create the things we see every day.
Some reallife examples of the lines and angles
(i) If you observe the bamboo plants, they grow in a straight line. Each bamboo stick resembles a straight line.
(ii) When we observe the railway tracks, they run in straight lines extending on both directions.
(iii) If we observe, the adjacent sides of the laptop, they form an angle between them.
(iv) The two legs of a ladder resemble straight lines. The two legs join at one point and the opening between them is the angle formed by these legs.
(v) What happens when you open a door?
Door
As soon as you open the door an angle is formed between the edge of the door and the threshold of the panel.
Formation of Angle
In the above figure, two raysandare the arm of an angle that meet at a common initial point Q (vertex)and form a ∠PQR. The measure of the angle PQR is written as ∠PQR but instead of writing this, we can simply write it as ∠PQR.
Example: Clock
The hourhand and the minutehand of the clock form a pair of complementary angles.
Complementary Angles
Here,
∠PQS + ∠SQR= 50° + 40° = 90°
In the above figure, we see that the sum of two angles is 90°.
Hence, ∠PQS and ∠SQR are complementary angles. And ∠PQS and ∠SQR are said to be complements of each other.
Example: Are the given angles complementary?
In the given figure,
∠AOB = 70°and ∠POQ = 20°
∠AOB + ∠POQ = 70° + 20° = 90°
Therefore, ∠AOB and ∠POQ are complementary angles.
Example: Clock
Supplementary Angle
Here,
∠PQS + ∠SQR = 150° + 30° = 180°
In the above figure, we see that the sum of two angles is 180°.
Hence, ∠PQS and ∠SQR are supplementary angles, and ∠PQS and ∠SQR are said to be supplements of each other.
Example: The following ∠AOB and ∠POQ are supplementary angles or not?
Supplementary Angle
Sol: In the given figure,
∠AOB + ∠POQ = 130° + 50° = 180°
∠AOB + ∠POQ = 180°
∴ ∠AOB and ∠POQ are supplementary angles. Or
∠AOB and ∠POQ are said to be supplements of each other.
These angles are such that:
(i) they have a common vertex.
(ii) they have a common arm.
(iii) the noncommon arms are on either side of the common arm.
Such pairs of angles are called adjacent angles.
Note: Adjacent angles have a common vertex and a common arm but no common interior points.
Example: In the following figure angles marked with 1 and 2 are they adjacent? If not give a reason for that.
Adjacent Angle
Solution:
In figure number (i)
(i) We see ∠PQS and ∠SQR have a common arm QS.
They have a common vertex Q.
They do not have a common interior point.
Hence, ∠PQS and ∠SQR are adjacent angles
In figure number (ii)
(ii) We see ∠RQS and ∠SQP have a common arm QS.
They have a common vertex Q.
They do not have a common interior point
Hence, ∠RQS and ∠SQP are adjacent angles
In figure number (iii)
(iii) We see ∠PTS and ∠SQR have a common arm QS.
They do not have a common vertex.
They do not have a common interior point.
The above figure does not satisfy all the conditions for being adjacent angles.
Hence, ∠PTS and ∠SQR are not adjacent angles.
Two lines are said to be intersecting when they cross each other at one point only and the point at which they intersect is called the point of intersection.
Here, two lines l and m intersect each other at point O, and point O is called the point of intersection.
A line that intersects two or more lines in a plane at distinct points is called a transversal line.
Here, the two lines l and m are intersected by a transversal n at points O and P respectively. We see that four angles are formed at each point O and P, namely ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, and ∠8.
Example: In the figure given below, l  m and ∠1 = 53°. Find ∠6 and ∠7.
We have,
∠1 = ∠3 [Vertically opposite angles]
∠3 = ∠7 [Corresponding angles]
∴ ∠1 = ∠7
→ ∠7 = 53° ∵ ∠1 = 53° (Given)
∠6 + ∠7 = 180° [Linear pair]
∠6 + 53° = 180°
∠6 + 53° − 53° = 180° − 53°
∠6 = 127°
Thus, ∠6 = 127° and ∠7 = 53°
If two lines lying in the same plane do not intersect when produced on either side, then such lines are said to be parallel to each other.
Here, lines l and m are parallel to each other, and transversal n intersects line l and m at point O and P respectively.
When the two parallel lines l and m are cut by a transversal n, then obtained the following relations:
(i) When a transversal intersects two parallel lines, then each pair of alternate interior angles are equal.
When line n intersects two parallel lines l and m, then we see that each pair of alternate interior angles is equal.
∴ ∠ 3 = ∠5, ∠4 = ∠6
(ii) When a transversal intersects two parallel lines, each pair of alternate exterior angles are equal.
When line n intersects two parallel lines l and m, then we see that each pair of alternate exterior angles is equal.
∴ ∠ 2 = ∠8, ∠1 = ∠7
(iii) When a transversal intersects two parallel lines, each pair of corresponding angles are equal.
When line n intersects two parallel lines l and m, then we see that each pair of corresponding angles is equal.
∴ ∠3 = ∠7, ∠2 = ∠6, ∠ 1 = ∠5, and ∠4 = ∠8
(iv) When a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal are supplementary
When line n intersects two parallel lines l and m, then we see that each pair of interior angles on the same side of the transversal are supplementary.
In the above figure, ∠ 3 = ∠4 .....Linear pair of angles
We know sum of the linear pair of angles is 180° ∴ ∠ 3 + ∠4 = 180°
But, ∠4 = ∠6 ...Pair of alternate interior angles Therefore, we can say that ∴ ∠ 3 + ∠6 = 180°
Similarly, ∠4 + ∠5 = 180°
Example: In the given figure l  m, ∠1 = 55°. Find ∠5, ∠6, and ∠7.
We have,
∠ 1 = ∠5 .....Corresponding angles
∴ ∠ 5 = 55° [∵ ∠1 = 55°]
∠5 = ∠7 .....Vertically opposite angles
∴ ∠7 = 55° [∵ ∠5 = 55°]
Now,
∠ 6 + ∠7 = 180°.....Linear pair of angles
∠ 6 + 55° = 180°
∠ 6 + 55° − 55° = 180° − 55°
∠ 6 = 180° − 55°
∠ 6 = 125°
Thus, ∠ 5 = 55°, ∠ 6 = 125°and ∠7 = 55°
Some special pairs of angles can be used to test if the lines are parallel or not.
(i) When a transversal intersects two parallel lines, such that if any pair of corresponding angles are equal, then the lines are parallel.
In the given figure, transversal n intersects two lines l and m in such a way that,
∠3 = ∠7, ∠2 = ∠6, ∠ 1 = ∠5, and ∠4 = ∠8 ...(Pairs of corresponding angles are equal)
Hence, we can say that lines are parallel.
(ii) When a transversal intersects two parallel lines, such that if any pair of alternate interior angles are equal, the lines have to be parallel.
In the given figure, transversal n intersects two lines l and m in such a way that,
∠ 3 = ∠5, ∠4 = ∠6 ... (Alternate interior angles are equal)
Hence, we can say that lines are parallel.
(iii) When transversal intersects two parallel lines, such that if any pair of alternate exterior angles are equal, the lines have to be parallel.
In the given figure, transversal n intersects two lines l and m in such a way that,
∠1 = ∠7, ∠2 = ∠8 ... (Alternate exterior angles are equal)
Hence, we can say that lines are parallel.
(v) When transversal intersects two parallel lines, such that if any pair of interior angles on the same side of the transversal are supplementary, the lines have to be parallel.
In the given figure, transversal n intersects two lines l and m in such a way that, ∠ 3, ∠6 and ∠4, ∠5 ...Pairs of cointerior angles or angles on the same sides of the transversal
Hence,
∠ 3 + ∠6 = 180° and
∠4 + ∠5 = 180°
Hence, we can say that lines are parallel.
Example: Find whether AB  CD.
In the given figure,
∠CPN = ∠OPD = 65°...Vertically opposite angles
∠BOP + ∠OPD = 180°
Thus, the sum of cointerior angle is 180°
Hence,
135° + ∠OPD = 180°
∠OPD = 180° − 135°
∠OPD = 45°
Therefore, AB  CD
76 videos345 docs39 tests

1. What is the definition of an angle in geometry? 
2. What are related angles in geometry? 
3. How do you determine if two lines are parallel to each other? 
4. What are some common pairs of lines in geometry? 
5. How can you check if two lines are perpendicular to each other? 

Explore Courses for Class 7 exam
