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Class 7 Maths Chapter 5 Notes Parallel and Intersecting Lines
Class 7 Maths Notes Chapter 5 – Class 7 Parallel and Intersecting Lines Notes
→ When two lines intersect, the vertically opposite angles are equal, and the sum of linear angles is 180°.
→ When two lines intersect such that all angles are equal, i.e., each angle is 90°, then the lines are said to be perpendicular lines.
→ When two lines in a plane do not intersect, they are called parallel lines.
→ When a transversal intersects a pair of parallel lines, the corresponding angles are always equal.
→ When a transversal intersects a pair of lines so that the corresponding angles are equal, then the pair of lines is always parallel.
→ When a transversal intersects a pair of parallel lines, the alternate angles are always equal.
→ When a transversal intersects a pair of parallel lines, the sum of the interior angles is always 180°.
Across the Line Class 7 Notes
1. Plane Surface
A flat surface that can be extended in all possible directions is called a plane surface.
Flat surfaces of cubes, cuboids, floors, tabletops, and blackboards are some examples of plane surfaces.
In this chapter, we shall discuss the relationship between lines lying on a plane surface.
2. Intersecting Lines
Two lines lying on a plane surface are called intersecting lines if they have exactly one common point. The common point is called the point of intersection of the lines.
In the figure, l and m are intersecting lines, and P is their point of intersection.
It may be noted that two lines cannot intersect at more than one point.
3. Angles Between Intersecting Lines
In this section, we shall discuss the angles made by a pair of intersecting lines.
In the figure, l and m are intersecting lines. At the point of intersection P, the lines l and m make four angles ∠a, ∠b, ∠c, and ∠d.
Using a protractor, measure angle ∠a. We find ∠a is equal to 55°.
In the figure, ∠a and ∠b are a linear pair of angles, and hence their sum must be 180°.
∴ ∠a + ∠b = 180°
⇒ 55° + ∠b = 180°
⇒ ∠b = 180° – 55°
⇒ ∠b = 125°
∠b and ∠c are also a linear pair of angles.
∴ ∠b + ∠c = 180°
⇒ 125° + ∠c = 180°
⇒ ∠c = 180° – 125°
⇒ ∠c = 55°
∠c and ∠d are also a linear pair of angles.
∴ ∠c + ∠d = 180°
⇒ 55° + ∠d = 180°
⇒ ∠d = 180° – 55°
⇒ ∠d = 125°
Also, using a protractor, we find that ∠b = 125°, ∠c = 55°, and ∠d = 125°.
Thus, we see that opposite angles ∠a and ∠c are equal.
Opposite angles ∠b and ∠d are also equal.
4. Vertically Opposite Angles
Let l and m be two lines intersecting at point P. At P, lines l and m make angles ∠a, ∠b, ∠c, and ∠d.
In the figure, ∠a and ∠b are a linear pair of angles.
∴ ∠a + ∠b = 180°
Also, ∠a and ∠d are a linear pair of angles.
∴ ∠a + ∠d = 180°
∴ ∠a + ∠b = ∠a + ∠d or ∠b = ∠d
∴ The opposite angles ∠b and ∠d are always equal.
Similarly, opposite angles ∠a and ∠c are always equal.
The opposite angles ∠b and ∠d are called vertically opposite angles.
∴ Thus, vertically opposite angles are always equal.
∠a and ∠c are also vertically opposite angles.
We conclude that when two lines intersect at a point, we find two pairs of vertically opposite angles, which are equal.
5. Measurements and Geometry
We know that vertically opposite angles are always equal.
In the figure, lines l and m intersect at point P. ∠a and ∠b are vertically opposite angles. When we measure ∠a and ∠b by using a protractor, we generally find that ∠a and ∠b are not exactly equal, and we may find a slight difference in our protractor readings. This happens due to the following reasons:
- The calibration of the protractor might not be perfect.
- Measuring error caused by the observer.
- The thickness of the lines drawn.
By definition, a line in geometry does not have any thickness, and it is impossible to draw a line on paper without thickness.
Perpendicular Lines Class 7 Notes
Two intersecting lines are called perpendicular lines if all four angles of intersection are equal.
In the figure, lines l and m intersect at P, and the angles of intersection ∠a, ∠b, ∠c, and ∠d are all equal.
∠a and ∠b are a linear pair of angles.
∴ ∠a + ∠b = 180°
⇒ ∠a + ∠a = 180°
⇒ 2∠a = 180°
⇒ ∠a = 90°
∴ ∠b = 180° – ∠a
⇒ ∠b = 180° – 90°
⇒ ∠b = 90°
∴ ∠b = ∠d (vertically opposite angles)
∴ ∠d = 90°
∴ ∠c = ∠a = 90° (vertically opposite angles)
Alternatively, we can also define perpendicular lines as follows:
Two intersecting lines are called perpendicular lines if they intersect each other at a right angle (90°).
Notation
Perpendicular lines are indicated with a square angle between them.
In the figure, lines l and m are perpendicular lines.
Between Lines Class 7 Notes
Consider the following pairs of lines.
Lines AB and A’B’ are intersecting lines, and P is their point of intersection.
Lines CD and C’D’ are in the same plane and do not intersect. On extending these lines to the left, they are going to intersect each other.
Lines EF and E’F’ are in the same plane and do not intersect. Moreover, they are not going to intersect each other even after extending these lines to any side. We are interested in this type of pair of lines.
1. Parallel Lines
Two lines lying in the same plane are called parallel lines if they do not meet each other, however far we extend them from both ends. In the figure, l and m are parallel lines.
2. Notation
We use an arrow mark to denote a set of parallel lines. If there is more than one set of parallel lines, the second set is shown by using two arrow marks, and so on.
In the above figure, lines l and m are parallel lines. Also, lines l’ and m’ are parallel lines.
Question 1.
Name some parallel lines you can spot in your classroom.
Solution:
In our classroom:
- The vertical sides of the blackboard are parallel to each other.
- The horizontal sides of the blackboard are parallel to each other.
- The opposite sides of the table are parallel to each other.
- The vertical sides of doors and windows are parallel to each other.
- Horizontal sides of doors and windows are parallel to each other.
Question 2.
Which pairs of lines appear to be parallel in the given figure?
Solution:
In the given figure:
- Lines a, i, and h are parallel to each other.
- Lines b and e are parallel to each other.
- Lines c and g are parallel to each other.
- Lines d and f are parallel to each other.
Parallel and Perpendicular Lines in Paper Folding Class 7 Notes
In this section, we shall learn the method of drawing parallel lines and perpendicular lines using the paper-folding method.
Activity 1: Draw a set of three parallel lines on a rectangular paper.
Step 1: Take a rectangular sheet of paper and mark its vertices as A, B, C, and D. (Figure)
Step 2: Fold the paper so that the point D overlaps the point A and the point C overlaps the point B, and press to form a crease. (Figure)
Step 3: Fold the folded paper in the same way once again and press to form a crease. (Figure)
Step 4: Unfold the folded rectangle ABCD and draw lines l, m, and n along the three creases. (Figure)
Step 5: l, m, n is the required set of three parallel lines on the given rectangular paper. (Figure)
Activity 2: Draw a set of three perpendicular lines to a line on a rectangular paper.
Step 1: Take a rectangular sheet of paper and mark its vertices as A, B, C, and D. (Figure)
Step 2: Fold the paper so that the point D overlaps the point A and the point C overlaps the point B, and press to form a crease. (Figure)
Step 3: Unfold the folded rectangle ABCD and draw line l along the crease. (Figure)
Step 4: Fold the rectangle ABCD in Figure. so that point D overlaps point C, and point A overlaps point B. (Figure)
Step 5: Fold the folded paper in the same way once again and press to form a crease. (Figure)
Step 6: Unfold the folded rectangle ABCD and draw lines m, n, and p along the three creases. (Figure)
Step 7: m, n, p is the required set of three perpendicular lines to the line l on the given rectangular paper. (Figure)
Transversals Class 7 Notes
A line is called a transversal if it intersects two different lines in the same plane.
In the figure, t is a transversal and it intersects two lines l and m lying in the same plane.
The transversal t makes eight angles with lines l and m.
Since vertically opposite angles are always equal, we have ∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7, and ∠6 = ∠8.
∴ Distinct angles between transversal t and lines l and m are ∠1, ∠2, ∠5, and ∠6.
∴ The transversal t makes 8 angles with lines l and m, and a maximum of four angles can have distinct measures.
Corresponding Angles Class 7 Notes
Let a transversal t intersect two lines l and m. The lines may or may not be parallel to each other. ∠1 and ∠5 are called corresponding angles.
Also, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 are corresponding angles. Corresponding angles may or may not be equal. Now we shall discuss the condition on the lines l and m so that the corresponding angles may be equal.
Activity 1: Draw a pair of lines and a transversal such that they form only two distinct angles.
We know that when a transversal intersects two lines, it forms eight angles and with a maximum of four distinct angle measures.
Here, we shall draw a pair of lines and a transversal with exactly two distinct angle measures.
Step 1: Draw a transversal t intersecting line l at points. (Figure)
Step 2: Using a protractor, measure angle ∠a.
We have ∠a equal of 55°.
∠a and ∠b are a linear pair of angles.
∴ ∠a + ∠b = 180°
⇒ 55° + ∠b = 180°
⇒ ∠b = 180° – 55°
⇒ ∠b = 125°
Also, ∠c = ∠b
∴ ∠c = 125°
Also, ∠d = ∠a
∴ ∠d = 55°
∴ We have two distinct angles, 55° and 125°.
Step 3: We shall draw another line m intersecting the transversal t, so that we do not get any new distinct angle.
Step 4: Take a point B on the transversal t. Using a protractor, draw a line m passing through B, so that it makes ∠55° with the transversal t. (Figure)
Step 5: We can easily calculate that: ∠f = 125°, ∠g = 55°, and ∠h = 125°.
Step 6: l and m are a pair of lines, and the transversal t makes only two distinct angles 55° and 125°. Here, lines l and m are parallel to each other.
Remark: In the above activity, the corresponding angles ∠a and ∠e are equal and each is equal to 55°. Similarly, other pairs of corresponding angles are equal.
∴ When the corresponding angles formed by a transversal on a pair of lines are equal to each other, the pair of lines is parallel to each other.
Activity 2: When a transversal intersects two parallel lines, the corresponding angles are always equal to each other.
Step 1: Let l and m be the given parallel lines. (Figure)
Step 2: Draw a transversal t intersecting the lines l and m at A and B, respectively. (Figure)
Step 3: In Figure 2, ∠a and ∠e are corresponding angles. Using a protractor, find ∠a and ∠e.
We have ∠a = 63° and ∠e = 63°
∴ The corresponding angles ∠a and ∠e are equal.
Similarly, other pairs of corresponding angles ∠b and ∠f, ∠c and ∠g, ∠d and ∠h are also equal.
Step 4: We conclude that when a transversal intersects two parallel lines, the corresponding angles are always equal to each other.
∴ Corresponding angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
Activity 3: When a pair of lines is not parallel to each other, the corresponding angles formed by a transversal can never be equal to each other.
Step 1: Let l and m be a pair of lines that are not parallel to each other.
Let t be a transversal intersecting l and m at A and B, respectively. (Figure)
Step 2: In Figure 1, ∠b and ∠f are corresponding angles. We measure ∠b and ∠f using a protractor. We have ∠b = 62° and ∠f = 40°.
∴ Corresponding angles ∠b and ∠f are not equal.
Step 3: Using ∠b = 62°, we have
∠a = 180° – b° = 180° – 62° = 118°.
∠c = 180° – b° = 180° – 62° = 118°.
∠d = 180° – c° = 180° – 118° = 62°.
Using ∠f = 40°, we have ∠g = 140°, ∠h = 40° and ∠e = 140°.
Step 4: We have ∠c ≠ ∠g, ∠d ≠ ∠h, and ∠a ≠ ∠e.
Step 5: Thus, other pairs of corresponding angles are not equal to each other.
Step 6: We find that when a pair of lines is not parallel to each other, then the corresponding angles formed by a transversal can never be equal to each other.
1. Recognising Corresponding Angles (‘Angles in shape F’)
It is interesting to note that the uppercase letter ‘F’ in the alphabet has corresponding angles.
This facilitates recognising corresponding angles when a transversal intersects two lines.
The following figures show the position of corresponding angles using the letter ‘F’.
Corresponding angles (F-angles)
Drawing Parallel Lines Class 7 Notes
In this section, we shall learn the method of drawing parallel lines with the help of a ruler and a set square.
Step 1: Draw a line using a ruler. (Figure)
Step 2: Place the ruler along the line and place a set square along the line so that its smaller side touches the ruler. Draw a line l’ along the other smaller side of the set square. (Figure)
Step 3: Slide the set square to the left side along the ruler. Draw another line l” along the other smaller side of the set square. (Figure)
Step 4: The angle between the smaller sides of a set square is always 90°.
∴ ∠a = 90° and ∠b = 90°
∴ ∠a = ∠b
Here, the transversal l intersects two lines l’ and l”.
Also, ∠a and ∠b are corresponding angles.
Since ∠a and ∠b are equal, the lines l’ and l” must be parallel lines.
Step 5: In the Figure, we have drawn parallel lines l’ and l” by using a ruler and a set square.
Making Parallel Lines Through a Given Point Using Paper Folding
In this section, we shall learn the method of drawing a line parallel to a given line and also passing through a given point.
Step 1: Draw a line l and a point P on a rectangular piece of paper. (Figure)
Step 2: Fold the paper so that the line l overlaps itself and its crease passes through the point P. Draw line l’ along this crease. (Figure)
Step 3: Fold the paper again so that the line l’ overlaps itself and it passes through the point P. Draw line l” along this crease. (Figure)
Step 4: We have ∠a = 90° and ∠b = 90°.
∴ ∠a = ∠b
Since corresponding angles ∠a and ∠b are equal, the lines l and l” are parallel lines.
Step 5: Line l” passes through the given point P and is also parallel to the given line l.
Alternate Angles Class 7 Notes
Let a transversal t intersect two parallel lines l and m.
∠3 and ∠5 are called alternate angles.
Also, ∠4 and ∠6 are alternate angles. ∠3 is equal to ∠7, corresponding angles.
∠7 is equal to ∠5, being vertically opposite angles.
∴ ∠3 is equal to ∠5.
∴ Alternate angles ∠3 and Z5 are equal to each other.
Similarly, alternate angles ∠4 and ∠6 are equal to each other.
∴ Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
1. Z-Angles
It is interesting to note that the uppercase letter ‘Z’ in the alphabet has alternate angles.
This facilitates recognising alternate angles when a transversal intersects two parallel lines.
The following figure shows the position of alternate angles using the letter ‘Z’.
Alternate angles (Z-angles)
2. Interior Angles
Let a transversal t intersect two parallel lines l and m. ∠4 and ∠5 are called interior angles.
Also, ∠3 and ∠6 are interior angles.
∠4 is equal to ∠8, corresponding angles.
∠8 and ∠5 are a pair of linear angles.
∴ ∠8 + ∠5 = 180°
∴ ∠4 + ∠5 = 180°
∴ The sum of interior angles ∠4 and ∠5 is 180°.
Similarly, the sum of the interior angles ∠3 and ∠6 is 180°.
∴ The sum of interior angles formed by a transversal intersecting a pair of parallel lines is always 180°.
Question 1.
In the given figure, lines l and m are parallel to each other, intersected by the transversal t.
If ∠6 is 135°, what are the measures of the other angles?
Solution:
We have ∠6 = 135°
∠8 = 135°, because ∠6 and ∠8 are vertically opposite angles.
∠7 = 180° – 135° = 45°, because ∠6 and ∠7 are a pair of linear angles.
∠5 = 45°, because ∠5 and ∠7 are vertically opposite angles.
∠1 = 45°, because ∠1 and ∠5 are corresponding angles.
∠2 = 135°, because ∠2 and ∠6 are corresponding angles.
∠3 = 45°, because ∠3 and ∠7 are corresponding angles.
∠4 = 135°, because ∠4 and ∠8 are corresponding angles.
Question 2.
In the given figure, lines l and m are intersected by the transversal t. If ∠a is 120° and ∠f is 70°, are lines l and m parallel to each other?
Solution:
We have ∠a = 120° and ∠f = 70°.
∴ ∠b = 180° – 120° = 60°, because ∠a and ∠b are a pair of linear angles.
∠b and ∠f are corresponding angles.
These angles are not equal, because ∠b = 60° and ∠f = 70°.
∴ Lines l and m cannot be parallel lines.
Question 3.
In the given figure, lines l and m are parallel to each other and intersected by the transversal t. If ∠3 = 50°, what is the measure of ∠6?
Solution:
We have ∠3 = 50°
∴ ∠2 = 180° – 50° = 130°, because ∠2 and ∠3 are a pair of linear angles.
∠6 = ∠2, because ∠2 and ∠6 are corresponding angles.
∴ ∠6 = 130°
Question 4.
In the given figure, line segment AB is parallel to DC, and AD is parallel to BC. ∠DAC is 65° and ∠ADC is 60°. What are the measures of angles ∠CAB, ∠ABC, and ∠BCD?
Solution:
∠BCD = 180° – 60° = 120°, because ∠ADC and ∠BCD are interior angles.
∠ACB = 65°, because ∠ACB and ∠DAC (= 65°) are alternate angles.
We have ∠BCD = ∠ACB + ∠ACD
⇒ 120° = 65° + ∠ACD
⇒ ∠ACD = 120° – 65°
⇒ ∠ACD = 55°
∴ ∠CAB = 55°, because ∠CAB and ∠ACD are alternate angles.
∴ ∠ABC = 180° – 120° = 60°, because ∠ABC and ∠BCD are interior angles.
∴ ∠CAB = 55°, ∠ABC = 60° and ∠BCD = 120°.
Parallel Illusion Class 7 Notes
There do not seem to be any parallel lines here. Or, are there?
What causes these illusions?
Solution:
(i) In the given figure, only two vertical lines, l and m, are parallel to each other.
(ii) In the given figure, all horizontal small lines are parallel to each other.
(iii) In the given figure, all horizontal small lines are parallel to each other. All vertical small lines are also parallel to each other.
(iv) In the given figure, only two horizontal lines, l and m, are parallel to each other.
