Lines And Angles

Hello my dear students! Welcome to today's mathematics lesson. I am so happy to see you all ready to learn something new and interesting. Today, we are going to study Chapter 6, which is all about Lines and Angles. This is a very important chapter that will help you understand the geometry around you. So let's begin our journey together.

So students, in the previous chapter, you learned that a minimum of two points are required to draw a line. You also studied some axioms and with the help of those axioms, you proved some other statements. In this chapter, we will go deeper into the world of geometry. We will study the properties of angles formed when two lines intersect each other, and also the properties of angles formed when a line intersects two or more parallel lines at distinct points. Further, we will use these properties to prove some statements using deductive reasoning. You have already verified many of these statements through some activities in your earlier classes, so this will be like revisiting old friends with a deeper understanding.

Now, let me ask you something. Have you ever looked around and noticed the different types of angles formed between the edges of surfaces in your daily life? Think about the corners of your classroom, the roof of a house, or the hands of a clock. All these involve angles. For making a model, say for example, you want to make a model of a hut for your school exhibition using bamboo sticks. How would you make it? You would keep some sticks parallel to each other, and some sticks would be slanted. Whenever an architect has to draw a plan for a multistoried building, she has to draw intersecting lines and parallel lines at different angles. Without the knowledge of the properties of these lines and angles, do you think she can draw the layout of the building? Of course not!

In science, you study the properties of light by drawing ray diagrams. For example, to study the refraction property of light when it enters from one medium to another, you use the properties of intersecting lines and parallel lines. When two or more forces act on a body, you draw a diagram in which forces are represented by directed line segments to study the net effect on the body. At that time, you need to know the relation between the angles when the rays or line segments are parallel to or intersect each other. To find the height of a tower or to find the distance of a ship from a lighthouse, you need to know the angle formed between the horizontal and the line of sight. There are so many examples where lines and angles are used. In the subsequent chapters of geometry, you will be using these properties of lines and angles to deduce more and more useful properties.

So now, let us first revise the terms and definitions related to lines and angles that you have learned in earlier classes.

So students, let us start with the basic terms. You all know what a line is, but let me remind you about some important definitions. A part or portion of a line with two end points is called a line-segment. For example, if you take two points A and B on a line, the portion between them is the line segment AB. We denote the line segment AB by the same symbol AB, and its length is also denoted by AB. Now, a part of a line with one end point is called a ray. For instance, if we start from point A and go infinitely in one direction, we get ray AB, denoted by AB. A line extends indefinitely in both directions. We denote a line by AB as well. However, the meaning will be clear from the context. Sometimes we also use small letters like l, m, n, and so on to denote lines.

Now, what do we mean by collinear and non-collinear points? If three or more points lie on the same line, they are called collinear points. Otherwise, they are called non-collinear points. For example, if you have three points A, B, and C all lying on a straight line, then they are collinear. But if you have three points that do not all lie on one single line, then they are non-collinear.

Now, let us talk about angles. You all know what an angle is, but let me explain it properly. An angle is formed when two rays originate from the same end point. The rays making an angle are called the arms of the angle, and the end point is called the vertex of the angle. For example, if two rays OA and OB start from point O, then angle AOB is formed, with O as the vertex and OA, OB as the arms.

Now, students, you have studied different types of angles in your earlier classes. Let me recall them with you. An acute angle is one that measures between 0° and 90°. For example, 30°, 45°, 60°, all these are acute angles. A right angle is exactly equal to 90°. The corners of your notebook or the angle between the floor and the wall of your classroom are right angles. An obtuse angle is greater than 90° but less than 180°. For example, 120°, 150° are obtuse angles. A straight angle is equal to 180°. When two rays point in exactly opposite directions, they form a straight angle. And finally, a reflex angle is greater than 180° but less than 360°. For example, 270° is a reflex angle.

Now, there are some special relationships between angles that you should remember. Two angles whose sum is 90° are called complementary angles. For instance, 30° and 60° are complementary because 30° + 60° = 90°. Two angles whose sum is 180° are called supplementary angles. For example, 70° and 110° are supplementary because 70° + 110° = 180°.

Now, let us understand what adjacent angles are. Two angles are called adjacent if they have a common vertex, a common arm, and their non-common arms are on different sides of the common arm. Let me explain this with an example. Look at the figure where we have angle ABD and angle DBC. They share the vertex B, they share the arm BD, and their non-common arms are BA and BC, which lie on different sides of the common arm BD. So, angle ABD and angle DBC are adjacent angles. We can write angle ABC equals angle ABD plus angle DBC. This is because when two angles are adjacent, their sum is always equal to the angle formed by the two non-common arms.

Now, let me tell you something important. Angle ABC and angle ABD are not adjacent angles. Why? Because their non-common arms BD and BC lie on the same side of the common arm BA. So, students, remember this point - for two angles to be adjacent, the non-common arms must be on different sides of the common arm.

Now, what happens when the non-common arms of two adjacent angles form a line? In that case, they are called a linear pair of angles. So, if you have two adjacent angles and their non-common arms lie along a straight line, then together they form a linear pair.

Now, let us recall vertically opposite angles. When two lines intersect each other, they form vertically opposite angles. For example, if two lines AB and CD intersect at point O, then we have two pairs of vertically opposite angles. One pair is angle AOD and angle BOC. Can you find the other pair? Yes, the other pair is angle AOC and angle BOD. We will learn more about these shortly.

Now, students, let us move on to our next topic, which is about intersecting lines and non-intersecting lines.

Imagine you draw two different lines PQ and RS on a paper. You can draw them in two different ways. Either they can cross each other, or they can never meet each other. When two lines cross each other, they are called intersecting lines. When they never meet, no matter how far you extend them, they are called parallel lines or non-intersecting lines. Remember, a line extends indefinitely in both directions. So, lines PQ and RS in one figure are intersecting lines, and in another figure, they are parallel lines.

Now, here is an interesting property of parallel lines. The length of the common perpendicular at different points on parallel lines is always the same. This equal length is called the distance between two parallel lines. Think about the railway tracks - the distance between the two parallel rails is constant throughout.

Now, students, let us learn about some important relations between angles. We already learned about complementary angles, supplementary angles, adjacent angles, and linear pair of angles. Now, let us find out what happens when a ray stands on a line.

Draw a figure in which a ray stands on a line. Name the line as AB and the ray as OC. What are the angles formed at point O? They are angle AOC, angle BOC, and angle AOB. Now, can we write angle AOC plus angle BOC equals angle AOB? Yes, we can! This is because angle AOC and angle BOC are adjacent angles, and their sum equals the angle formed by the two non-common arms, which is angle AOB.

Now, what is the measure of angle AOB? It is 180° because it is a straight angle - the two arms OA and OB form a straight line. So, from this, we can say that angle AOC plus angle BOC equals 180°.

Now, students, this gives us a very important axiom. Let me state it clearly.

Axiom 6.1 says: If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.

This is also called the linear pair axiom. When the sum of two adjacent angles is 180°, they are called a linear pair of angles.

Now, here is something interesting. In Axiom 6.1, it is given that a ray stands on a line, and from this given information, we have concluded that the sum of two adjacent angles so formed is 180°. Can we write this the other way? That is, can we take the conclusion as given and the given as conclusion? Let me explain.

The statement would become: If the sum of two adjacent angles is 180°, then a ray stands on a line, meaning the non-common arms form a line. This is the converse of the original statement. We call each statement as the converse of the other.

Now, we need to check whether this converse is true or not. Let us draw adjacent angles of different measures and check if the non-common arms always form a line. You will find that only in one particular case, both the non-common arms lie along a straight line. In that case, the sum of the two adjacent angles is exactly 180°. From this, we can conclude that the converse is also true. So, we can state it as another axiom.

Axiom 6.2 says: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.

So, students, Axiom 6.1 and Axiom 6.2 together are called the Linear Pair Axiom. This is a very important concept that we will use again and again in geometry.

Now, let us examine what happens when two lines intersect each other. You already know from your earlier classes that when two lines intersect, the vertically opposite angles are equal. But now, let us prove this result using the linear pair axiom.

Here is Theorem 6.1: If two lines intersect each other, then the vertically opposite angles are equal.

Let us prove this. Let AB and CD be two lines intersecting at O. They lead to two pairs of vertically opposite angles: angle AOC and angle BOD, and angle AOD and angle BOC.

We need to prove that angle AOC equals angle BOD, and angle AOD equals angle BOC.

Now, ray OA stands on line CD. Therefore, angle AOC plus angle AOD equals 180°. This is because they form a linear pair.

Similarly, can we write angle AOD plus angle BOD equals 180°? Yes, because ray OD stands on line AB, so angle AOD and angle BOD form a linear pair.

Now, from these two equations, we can write: angle AOC plus angle AOD equals angle AOD plus angle BOD.

This implies that angle AOC equals angle BOD. This is because if we subtract angle AOD from both sides, we get angle AOC equals angle BOD.

Similarly, it can be proved that angle AOD equals angle BOC.

So, students, this is the proof that vertically opposite angles are equal when two lines intersect. This is a very important theorem, and you should remember it.

Now, let us do some examples based on the linear pair axiom and this theorem.

Example 1: In a figure, lines PQ and RS intersect each other at point O. If angle POR : angle ROQ equals 5 : 7, find all the angles.

Let me solve this for you. Angle POR and angle ROQ form a linear pair, so their sum is 180°. It is given that the ratio of angle POR to angle ROQ is 5 : 7. So, let angle POR be 5x and angle ROQ be 7x. Then, 5x plus 7x equals 180°, which gives 12x equals 180°, so x equals 15°. Therefore, angle POR equals 5 times 15, which is 75°. And angle ROQ equals 7 times 15, which is 105°.

Now, angle POS is vertically opposite to angle ROQ, so angle POS equals angle ROQ, which is 105°. And angle SOQ is vertically opposite to angle POR, so angle SOQ equals angle POR, which is 75°.

So, students, you see how we use the linear pair axiom and the theorem about vertically opposite angles to solve such problems.

Example 2: In a figure, ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of angle POS and angle SOQ respectively. If angle POS equals x, find angle ROT.

Let me solve this. Ray OS stands on line POQ, so angle POS plus angle SOQ equals 180°. Given that angle POS equals x, we can write x plus angle SOQ equals 180°, so angle SOQ equals 180° minus x.

Now, ray OR bisects angle POS, so angle ROS equals half of angle POS, which is x/2.

Similarly, ray OT bisects angle SOQ, so angle SOT equals half of angle SOQ, which is half of (180° minus x), which equals 90° minus x/2.

Now, angle ROT equals angle ROS plus angle SOT, which is x/2 plus 90° minus x/2, which equals 90°.

So, students, angle ROT is always 90°, regardless of the value of x. This is a very interesting result!

Example 3: In a figure, OP, OQ, OR, and OS are four rays. Prove that angle POQ plus angle QOR plus angle SOR plus angle POS equals 360°.

This is a proof question. Let me explain the solution. In the figure, we need to produce any of the rays backwards to a point. Let us produce ray OQ backwards to a point T so that TOQ is a line.

Now, ray OP stands on line TOQ, so angle TOP plus angle POQ equals 180°. This is by the linear pair axiom.

Similarly, ray OS stands on line TOQ, so angle TOS plus angle SOQ equals 180°.

But angle SOQ equals angle SOR plus angle QOR. So, the second equation becomes angle TOS plus angle SOR plus angle QOR equals 180°.

Now, adding the first and third equations, we get angle TOP plus angle POQ plus angle TOS plus angle SOR plus angle QOR equals 360°.

But angle TOP plus angle TOS equals angle POS. So, substituting this, we get angle POQ plus angle QOR plus angle SOR plus angle POS equals 360°.

And that is exactly what we wanted to prove! So, students, the sum of all four angles around a point is always 360°.

Now, students, let us move on to our next important topic, which is about lines parallel to the same line.

Here is a question: If two lines are parallel to the same line, will they be parallel to each other? Let us check this.

Look at the figure where line m is parallel to line l, and line n is also parallel to line l. Now, let us draw a line t as a transversal for lines l, m, and n. It is given that line m is parallel to line l, and line n is parallel to line l.

Now, since line m is parallel to line l, the corresponding angles formed by transversal t are equal. So, angle 1 equals angle 2. Similarly, since line n is parallel to line l, angle 1 equals angle 3. Therefore, angle 2 equals angle 3.

But angle 2 and angle 3 are corresponding angles, and they are equal. Therefore, we can say that line m is parallel to line n.

This result can be stated as Theorem 6.6: Lines which are parallel to the same line are parallel to each other.

This property can be extended to more than two lines as well. If three or more lines are parallel to the same line, then they are all parallel to each other.

Now, let us solve some examples related to parallel lines.

Example 4: In a figure, if PQ is parallel to RS, angle MXQ equals 135°, and angle MYR equals 40°, find angle XMY.

Let me solve this. Here, we need to draw a line AB parallel to line PQ through point M. Now, AB is parallel to PQ, and PQ is parallel to RS. Therefore, AB is also parallel to RS.

Now, angle QXM plus angle XMB equals 180° because AB is parallel to PQ, and XM is a transversal. These are interior angles on the same side of the transversal.

But angle QXM equals 135°. So, 135° plus angle XMB equals 180°, which gives angle XMB equals 45°.

Now, angle BMY equals angle MYR because AB is parallel to RS, and these are alternate angles. So, angle BMY equals 40°.

Now, angle XMY equals angle XMB plus angle BMY, which is 45° plus 40°, which equals 85°.

So, students, angle XMY is 85°.

Example 5: If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

Let me explain the proof. In the figure, a transversal AD intersects two lines PQ and RS at points B and C respectively. Ray BE is the bisector of angle ABQ, and ray CG is the bisector of angle BCS. It is given that BE is parallel to CG.

We need to prove that PQ is parallel to RS.

It is given that ray BE is the bisector of angle ABQ. Therefore, angle ABE equals half of angle ABQ.

Similarly, ray CG is the bisector of angle BCS. Therefore, angle BCG equals half of angle BCS.

But BE is parallel to CG, and AD is the transversal. Therefore, angle ABE equals angle BCG. This is by the corresponding angles axiom.

Substituting the values, we get half of angle ABQ equals half of angle BCS, which means angle ABQ equals angle BCS.

But these are the corresponding angles formed by transversal AD with PQ and RS, and they are equal. Therefore, by the converse of the corresponding angles axiom, PQ is parallel to RS.

So, students, this is how we prove that if the bisectors of corresponding angles are parallel, then the two lines are parallel.

Example 6: In a figure, AB is parallel to CD, and CD is parallel to EF. Also, EA is perpendicular to AB. If angle BEF equals 55°, find the values of x, y, and z.

Let me solve this. Angle y plus 55° equals 180° because they are interior angles on the same side of the transversal ED. Therefore, y equals 180° minus 55°, which is 125°.

Now, x equals y because AB is parallel to CD, and these are corresponding angles. Therefore, x equals 125°.

Now, since AB is parallel to CD and CD is parallel to EF, therefore, AB is parallel to EF.

So, angle EAB plus angle FEA equals 180° because they are interior angles on the same side of the transversal EA. But angle EAB is 90° because EA is perpendicular to AB. So, 90° plus z plus 55° equals 180°, which gives z equals 35°.

So, students, x equals 125°, y equals 125°, and z equals 35°.

Now, students, we have covered all the important concepts in this chapter. Let me summarize what we have learned today.

In this chapter, you have studied the following important points:

First, if a ray stands on a line, then the sum of the two adjacent angles so formed is 180°, and vice-versa. This property is called the Linear Pair Axiom. This is a fundamental concept that we use very often in geometry.

Second, if two lines intersect each other, then the vertically opposite angles are equal. This is Theorem 6.1, and we proved it using the linear pair axiom.

Third, lines which are parallel to the same line are parallel to each other. This is Theorem 6.6. This property can be extended to more than two lines as well.

We also learned about various types of angles: acute angles, right angles, obtuse angles, straight angles, and reflex angles. We learned about complementary angles (whose sum is 90°) and supplementary angles (whose sum is 180°). We learned about adjacent angles and linear pairs. We learned about intersecting lines and parallel lines, and the distance between parallel lines.

We also solved several examples that illustrate how to apply these concepts and theorems to solve problems. These examples showed us how to use the linear pair axiom, the theorem about vertically opposite angles, and the properties of parallel lines to find unknown angles and prove various results.

Students, this chapter forms the foundation for many more topics in geometry that you will study in the future. The concepts of lines and angles are used in almost every branch of mathematics and in many real-life applications. So, make sure you understand these concepts thoroughly and practice as many problems as you can.

That brings us to the end of today's lesson. I hope you all understood everything clearly. Remember, geometry is all about understanding and visualizing. So, always draw diagrams and try to relate the concepts to real-life situations. Thank you for your attention, and see you in the next lesson!