Welcome dear students! Today we are going to learn about Lines and Angles from Class 7 Maths. You already know how to identify different lines, line segments and angles in a given shape. Let us look at Figure 5.1 together. In this figure, we see four different geometric shapes: a triangle, a quadrilateral, a pentagon, and a hexagon. Can you identify the different line segments and angles formed in them? Can you also identify whether the angles made are acute or obtuse or right?
Before we proceed, please complete this activity: List ten figures around you and identify the acute, obtuse and right angles found in them. Look at your desk, windows, books, and doors. Note down the angles you see and classify them.
Recall that a line segment has two end points. If we extend the two end points in either direction endlessly, we get a line. Thus, we can say that a line has no end points. On the other hand, recall that a ray has one end point, namely its starting point. For example, look at the figures given below in Figure 5.2. Figure 5.2 part one shows a line segment PQ. Figure 5.2 part two shows a line. And Figure 5.2 part three shows a ray. A line segment PQ is generally denoted by the symbol PQ. A line AB is denoted by the symbol AB with a straight line drawn above the letters. The ray OP is denoted by the symbol OP with a ray symbol, an arrow pointing to one side, drawn above the letters. Give some examples of line segments and rays from your daily life and discuss them with your friends.
[CHECKPOINT]
Again recall that an angle is formed when lines or line segments meet. In Figure 5.1, observe the corners. These corners are formed when two lines or line segments intersect at a point. For example, look at the figures given below in Figure 5.3. In Figure 5.3 part one, line segments AB and BC intersect at B to form angle ABC, and again line segments BC and AC intersect at C to form angle ACB and so on. Whereas, in Figure 5.3 part two, lines PQ and RS intersect at O to form four angles POS, SOQ, QOR and ROP. An angle ABC is represented by the symbol ∠ABC. Thus, in Figure 5.3 part one, the three angles formed are ∠ABC, ∠BCA and ∠BAC. And in Figure 5.3 part two, the four angles formed are ∠POS, ∠SOQ, ∠QOR and ∠POR. You have already studied how to classify the angles as acute, obtuse or right angle. Note: While referring to the measure of an angle ABC, we shall write ∠ABC simply as ∠ABC. The context will make it clear, whether we are referring to the angle or its measure.
[CHECKPOINT]
Now let us move on to section 5.2, which is about Related Angles. First, we will study Complementary Angles. When the sum of the measures of two angles is 90°, the angles are called complementary angles. Whenever two angles are complementary, each angle is said to be the complement of the other angle. In the diagram shown in Figure 5.4, two angles are placed side by side, measuring 30° and 60°. The 30° angle is the complement of the 60° angle and vice versa. Now, look at the visual exercises in Figure 5.4. Are these two angles complementary? For the first pair, check your textbook figure. If the angles add to 90°, the answer is Yes. For the second pair, if they do not add to 90°, the answer is No. Apply this same check to the third and fourth pairs.
Think, Discuss and Write: Can two acute angles be complement to each other? Yes, they can, as long as their sum is exactly 90°. Can two obtuse angles be complement to each other? No, because each obtuse angle is greater than 90°, so their sum will always be greater than 180°. Can two right angles be complement to each other? No, because 90° plus 90° equals 180°, not 90°.
[CHECKPOINT]
Let us try some practice questions from Try These. In Figure 5.5, four pairs of angles are shown. To identify which pairs are complementary, look at the measures given in your textbook for each figure. Add the two angles in each pair. If the sum is exactly 90°, they are complementary. Next, what is the measure of the complement of each of the following angles? For 45°, the complement is 90° minus 45°, which equals 45°. For 65°, it is 90° minus 65°, which equals 25°. For 41°, it is 90° minus 41°, which equals 49°. For 54°, it is 90° minus 54°, which equals 36°. Here is another problem: The difference in the measures of two complementary angles is 12°. Find the measures of the angles. Let the smaller angle be x. Then the larger angle is x plus 12°. Since they are complementary, x plus (x plus 12°) equals 90°. Therefore, 2x plus 12° equals 90°. Subtract 12° from both sides to get 2x equals 78°. Divide by 2 to get x equals 39°. So the smaller angle is 39° and the larger angle is 39° plus 12°, which is 51°.
[CHECKPOINT]
Now let us look at Supplementary Angles. Look at the pairs of angles in Figure 5.6. Do you notice that the sum of the measures of the angles in each of the above pairs comes out to be 180°? Such pairs of angles are called supplementary angles. When two angles are supplementary, each angle is said to be the supplement of the other. Think, Discuss and Write: Can two obtuse angles be supplementary? No, because each is greater than 90°, so their sum exceeds 180°. Can two acute angles be supplementary? No, because each is less than 90°, so their sum is less than 180°. Can two right angles be supplementary? Yes, because 90° plus 90° equals exactly 180°. In Figure 5.7, several pairs of angles are shown. Find the pairs of supplementary angles by checking each pair. Look at the degree measures in your book. If any two angles add up to 180°, they form a supplementary pair.
[CHECKPOINT]
Now we will solve Exercise 5.1 completely. Question 1 asks to find the complement of three given angles shown in the figures. Look at the angles in your textbook. For each, subtract the given measure from 90° to find the complement. Question 2 asks to find the supplement of the angles shown in the figures. Similarly, subtract each given measure from 180°. Question 3 asks to identify complementary and supplementary pairs. Let us solve all six parts. (i) 65° and 115°: sum is 180°, therefore supplementary. (ii) 63° and 27°: sum is 90°, therefore complementary. (iii) 112° and 68°: sum is 180°, therefore supplementary. (iv) 130° and 50°: sum is 180°, therefore supplementary. (v) 45° and 45°: sum is 90°, therefore complementary. (vi) 80° and 10°: sum is 90°, therefore complementary. Question 4: Find the angle equal to its complement. Let it be x. x plus x equals 90°, so 2x equals 90°, therefore x equals 45°. Question 5: Find the angle equal to its supplement. x plus x equals 180°, so 2x equals 180°, therefore x equals 90°.
[CHECKPOINT]
Question 6: ∠1 and ∠2 are supplementary. If ∠1 is decreased, ∠2 must increase by the exact same amount to keep the sum at 180°. Question 7: Can two angles be supplementary if both are acute? No. Obtuse? No. Right? Yes. Question 8: An angle is greater than 45°. Is its complement greater than, equal to, or less than 45°? It must be less than 45°, because together they must equal 90°. Question 9: Fill in the blanks. (i) If two angles are complementary, then the sum of their measures is 90°. (ii) If two angles are supplementary, then the sum of their measures is 180°. (iii) If two adjacent angles are supplementary, they form a linear pair. Question 10: In the adjoining figure, we see intersecting lines with a ray. Name the following pairs. (i) Obtuse vertically opposite angles: Look for the two obtuse angles directly across from each other at the intersection. (ii) Adjacent complementary angles: Look for two angles sharing a common vertex and arm that together make 90°. (iii) Equal supplementary angles: These are two right angles that form a straight line. (iv) Unequal supplementary angles: Look for an acute and an obtuse angle that form a straight line. (v) Adjacent angles that do not form a linear pair: These are angles that share a vertex and arm but their non-common arms do not form a straight line.
[CHECKPOINT]
Now we move to section 5.3, Pairs of Lines. First, Intersecting Lines. The blackboard on its stand, the letter Y made up of line segments and the grill door of a window in Figure 5.8, what do all these have in common? They are examples of intersecting lines. Two lines l and m intersect if they have a point in common. This common point O is their point of intersection. In Figure 5.9, lines AC and BE intersect at P. Lines AC and BC intersect at C. Lines AC and EC intersect at C. Try to find another ten pairs of intersecting line segments in your surroundings. Should any two lines or line segments necessarily intersect? No. Can you find two pairs of non intersecting line segments in the figure? Yes, look for segments that run side by side without meeting. Can two lines intersect in more than one point? No, two distinct straight lines can intersect at only one point. Try These: Find examples where lines intersect at right angles. Look at corners of a book or window frame. Find the measures of the angles at the vertices of an equilateral triangle. Each is 60°. Draw a rectangle. The angles at the four vertices are all 90°. If two lines intersect, do they always intersect at right angles? No, only if they are perpendicular.
[CHECKPOINT]
Next is Transversal. You might have seen a road crossing two or more roads or a railway line crossing several other lines in Figure 5.10. These give an idea of a transversal. A line that intersects two or more lines at distinct points is called a transversal. In Figure 5.11, line p is a transversal to the lines l and m. In Figure 5.12, line p is not a transversal, although it cuts two lines l and m. Why? Because it cuts them at the same point, not at distinct points. Now, let us study the angles made by a transversal. In Figure 5.13, you see lines l and m cut by transversal p. The eight angles marked 1 to 8 have their special names. Interior angles are ∠3, ∠4, ∠5, and ∠6. Exterior angles are ∠1, ∠2, ∠7, and ∠8. Pairs of Corresponding angles are ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8. Pairs of Alternate interior angles are ∠3 and ∠6, and ∠4 and ∠5. Pairs of Alternate exterior angles are ∠1 and ∠8, and ∠2 and ∠7. Pairs of interior angles on the same side of the transversal are ∠3 and ∠5, and ∠4 and ∠6.
[CHECKPOINT]
Note: Corresponding angles, like ∠1 and ∠5 in Figure 5.14, include different vertices, are on the same side of the transversal, and are in corresponding positions relative to the two lines. Alternate interior angles, like ∠3 and ∠6 in Figure 5.15, have different vertices, are on opposite sides of the transversal, and lie between the two lines. Try These: Suppose two lines are given. How many transversals can you draw for these lines? Infinitely many. If a line is a transversal to three lines, how many points of intersections are there? Three points. Try to identify a few transversals in your surroundings. Now, Transversal of Parallel Lines. Do you remember what parallel lines are? They are lines on a plane that do not meet anywhere. Can you identify parallel lines in Figure 5.16? Look for lines that stay the same distance apart, like railway tracks or the edges of a notebook page.
[CHECKPOINT]
Let us do an activity. Take a ruled sheet of paper. Draw two parallel lines l and m. Draw a transversal t to the lines l and m. Label ∠1 and ∠2 as shown in Figure 5.17 part one. Place a tracing paper over the figure. Trace the lines l, m and t. Slide the tracing paper along t, until l coincides with m. You find that ∠1 on the traced figure coincides with ∠2 of the original figure. In fact, you can see all the following results by similar tracing and sliding activity. ∠1 equals ∠2. ∠3 equals ∠4. ∠5 equals ∠6. ∠7 equals ∠8. This activity illustrates the following fact: If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure. We use this result to get another interesting result. Look at Figure 5.18. When t cuts the parallel lines l and m, we get ∠3 equals ∠7 because they are vertically opposite angles. But ∠7 equals ∠8 because they are corresponding angles. Therefore, ∠3 equals ∠8. You can similarly show that ∠1 equals ∠6. Thus, we have the following result: If two parallel lines are cut by a transversal, each pair of alternate interior angles are equal.
[CHECKPOINT]
This second result leads to another interesting property. Again, from Figure 5.18, ∠3 plus ∠1 equals 180° because ∠3 and ∠1 form a linear pair. But ∠1 equals ∠6, which is a pair of alternate interior angles. Therefore, we can say that ∠3 plus ∠6 equals 180°. Similarly, ∠1 plus ∠8 equals 180°. Thus, we obtain the following result: If two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are supplementary. You can very easily remember these results if you can look for relevant shapes. The F shape stands for corresponding angles. The Z shape stands for alternate angles. Do This: Draw a pair of parallel lines and a transversal. Verify the above three statements by actually measuring the angles with a protractor. Try These: In Figure 5.16, name the pairs of angles in each figure. Identify corresponding, alternate interior, and interior same-side angles for each transversal shown.
[CHECKPOINT]
Now we move to section 5.4, Checking for Parallel Lines. If two lines are parallel, then you know that a transversal gives rise to pairs of equal corresponding angles, equal alternate interior angles and interior angles on the same side of the transversal being supplementary. When two lines are given, is there any method to check if they are parallel or not? You need this skill in many life oriented situations. A draftsman uses a carpenter square and a straight edge ruler to draw these segments in Figure 5.19. He claims they are parallel. How? Are you able to see that he has kept the corresponding angles to be equal? What is the transversal here? The ruler acts as the transversal. Thus, when a transversal cuts two lines, such that pairs of corresponding angles are equal, then the lines have to be parallel. Look at the letter Z in Figure 5.20. The horizontal segments here are parallel, because the alternate angles are equal. When a transversal cuts two lines, such that pairs of alternate interior angles are equal, the lines have to be parallel.
[CHECKPOINT]
Draw a line l in Figure 5.21. Draw a line m, perpendicular to l. Again draw a line p, such that p is perpendicular to m. Thus, p is perpendicular to a perpendicular to l. You find p is parallel to l. How? This is because you draw p such that ∠1 plus ∠2 equals 180°. Thus, when a transversal cuts two lines, such that pairs of interior angles on the same side of the transversal are supplementary, the lines have to be parallel. Try These: Check if line l is parallel to line m by verifying if the corresponding or alternate angles are equal. If l is parallel to m, find angle x by using the supplementary or equal angle properties. Look at the figures in your book, apply the rules, and solve for the unknowns.
Now let us solve Exercise 5.2 completely. Question 1: State the property that is used in each statement. (i) If a is parallel to b, then ∠1 equals ∠5. This uses corresponding angles. (ii) If ∠4 equals ∠6, then a is parallel to b. This uses alternate interior angles. (iii) If ∠4 plus ∠5 equals 180°, then a is parallel to b. This uses interior angles on the same side being supplementary.
[CHECKPOINT]
Question 2: In the adjoining figure, identify the pairs. (i) Corresponding angles are ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8. (ii) Alternate interior angles are ∠3 and ∠6, and ∠4 and ∠5. (iii) Interior angles on the same side of the transversal are ∠3 and ∠5, and ∠4 and ∠6. (iv) Vertically opposite angles are ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, and ∠6 and ∠8. Question 3: In the adjoining figure, lines p and q are parallel, cut by a transversal. One angle is given as 125°. Using the linear pair, the adjacent angle is 180° minus 125°, which equals 55°. Using vertically opposite angles, the angle opposite 125° is 125°. Using corresponding angles, the angle corresponding to 125° is 125°, and the angle corresponding to 55° is 55°. All other angles are found using these relationships. Question 4: Find the value of x if l is parallel to m. In figure one, x and 110° are corresponding angles, so x equals 110°. In figure two, x and 100° are alternate interior angles, so x equals 100°.
[CHECKPOINT]
Question 5: In the given figure, the arms of two angles are parallel. If ∠ABC equals 70°, then find ∠DGC and ∠DEF. Since the arms are parallel, ∠DGC and ∠ABC are corresponding angles. Therefore, ∠DGC equals 70°. Similarly, ∠DEF and ∠DGC are corresponding angles. Therefore, ∠DEF equals 70°. Question 6: In the given figures, decide whether l is parallel to m. In figure (i), the interior angles on the same side are 126° and 54°. Their sum is 126° plus 54°, which equals 180°. Since they are supplementary, l is parallel to m. In figure (ii), the alternate interior angles are both 75°. Since they are equal, l is parallel to m. In figure (iii), the corresponding angles are 80° and 70°. Since they are not equal, l is not parallel to m. In figure (iv), the interior angles on the same side are 100° and 80°. Their sum is 180°, so l is parallel to m.
[CHECKPOINT]
Let us review what we have discussed. First, a line segment has two end points. A ray has only one end point, its initial point. A line has no end points on either side. Second, when two lines l and m meet, we say they intersect. The meeting point is called the point of intersection. When lines drawn on a sheet of paper do not meet, however far produced, we call them to be parallel lines. We also learned about complementary angles, which sum to 90°, and supplementary angles, which sum to 180°. We studied transversals cutting parallel lines and discovered three key properties: corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side of the transversal are supplementary. We also learned how to use these properties in reverse to check if two lines are parallel. Practice identifying these angle pairs in your surroundings and in your textbook diagrams.
[CHECKPOINT]
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]