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SCERT Class 7 Maths Chapter 1 Solutions Parallel Lines
Class 7 Maths Chapter 1 Parallel Lines Questions and Answers Kerala State Syllabus
Parallel Lines Class 7 Questions and Answers Kerala Syllabus
Page 17
Question 1.
Draw the parallelogram below with the given measures.
Calculate the other three angles.
Answer:
Draw a horizontal line segment AB that is 5 cm long. This will be one side of the parallelogram.
At point A, use a protractor to measure and mark an angle of 60 degrees from line AB.
From point A, draw a line segment AC that is 3 cm long, making sure it forms the 60° angle with AB.
From point B, draw a line segment BD that is 3 cm long, parallel to AC. You can use a ruler and a set square to ensure parallelism.
From point C, draw a line segment CD that is 5 cm long, parallel to AB. The points C and D should connect to complete the parallelogram.
Now finding the remaining angles,
We know that, ∠A + ∠C = 180°
∠C = 180° – 60° = 120°
Similarly, ∠A + ∠B = 180°
∠B = 180° – 60° = 120°
And, ∠B + ∠D = 180°
∠D = 180° – 120° = 60°
Question 2.
The top and bottom blue lines in the figure are parallel. Find the angle between the green lines.
Answer:
Draw a horizontal line parallel to the other two parallel lines
From the figure, we get,
∠EAB = ∠ABD = 40°
∠DBC – ∠BGF = 50°
:. ∠ABC = ∠ABD + ∠DBC
= 40° + 50° = 90°
Question 3.
In the figure, the pair of lines slanted to the left are parallel; and also the pair of lines slanted to the right.
Draw this figure:
Answer:
Draw a line segment AB measuring 2 cm.
Draw a perpendicular line from point A.
Measure 20° on the protector, draw two lines on both sides of the perpendicular line, and mark C and D.
Using a set square, draw a line parallel to line AC starting from point B
Using a set square, draw a line parallel to line AD starting from point B and mark the intersection point as E.
Page 21
Question 1.
Draw the triangle with the given measures.
Answer:
First off all we have to find the third angle of the given triangle
⇒ Third angle = 180° – (60° + 40°) = 180° – 100° = 80°
Now we can start drawing the triangle, for that
Draw a horizontal line segment that is 5 cm long.
At left end of the line, use a protractor to measure and mark an angle of 40° from the line
At right end of the line, use a protractor to measure and mark an angle of 40°from the line
Construct line segments forming angles of 40° and 80°, extending each until the lines intersect
Question 2.
The figure shows a triangle drawn in a rectangle. Calculate the angles of the triangle.
Answer:
Consider the figure
Since, ABCD is rectangle ∠A, ∠B, ∠C, ∠D equals to 90°
Now, it is given that one part of ∠A is 40°, so the remaining part of A is 50°
Similarly, one part of ∠D is 25°, so the remaining part of ∠D is 65°
It implies that two angles of the triangle are 50° and 65°.
Therefore, the third angle of the triangle is = 180° – (50° + 65°)
= 180° – (105°)
= 75°
So, the angles of triangles are 50°, 65°, 75o
Question 3.
The top and bottom lines in the figure are parallel. Calculate the third angle of the bottom triangle and all angles of the top triangle.
Answer:
Consider the above figure,
Given that AB and CD are parallel lines
The two given angles of the bottom triangle are 35° and 45°
Therefore, third angle of bottom triangle = 180° – (35° +45°)
= 180° – (80°)
= 100°
In bottom triangle, ∠B and in top triangle, ∠C are Alternate angles
Therefore, ∠A and ∠D are equal. ∠D of top triangle is = 35°
Similarly, in bottom triangle, ∠B and in top triangle, ∠C are Alternate angles
Therefore, B and C are equal, ∠C of top triangle is = 45°
So, the third angle of top triangle = 180° – (35° + 45o)
= 180° – (80°)
= 100°
All angles of bottom triangle = 35°, 45°,
All angles of top triangle = 35°, 45°, 100°.
Question 4.
The left and right sides of the large triangle are parallel to the left and right sides of the small triangle. Calculate the other two angles of the large triangle and all angles of the small triangle.
Answer:
Consider the above figure,
Given that AB and CD are parallel, and AC and DE are also parallel,
It’s given that the middle angle is 70°.
Considering the parallel lines AB and CD, AC is a slanting line intersecting the parallel lines.
Therefore, the middle angle and ∠A of the larger triangle are alternate angles,
Therefore, these two angles are equal
i.e., ∠A = 70°
Now, we can calculate the third angle of the larger triangle
i.e., ∠B = 180° – (70° +60°)
= 180° – (130°) = 50°
All three angles of larger triangle are 50°, 60°, 70°
Similarly, considering the parallel lines AC and DE, CD is a slanting line intersecting the parallel lines.
Considering the parallel lines AC and DE, CD is slanting line through the parallel lines.
Here, middle angle and the ∠D of smaller triangle are alternate angles,
Therefore, these two angles are equal
i.e., ∠D = 70°
Now, ZC of larger triangle, middle angle.and ∠C of the smaller triangle form a straight line.
Therefore, ∠C of smaller triangle is = 180° − (60° + 70°)
= 180° – 130°
= 50°
Now we can find the third angle of smaller triangle (∠E) = 180° – (50° + 70°)
= 180° – (120°)
= 60°
All three angle of smaller triangle are 50°, 60°, 70°
Question 5.
A triangle is drawn inside a parallelogram.
Calculate the angles of the triangle.
Answer:
Consider the above figure,
∠D and ∠B are opposite angles of parallelogram.
Therefore, D = ∠B = 110°
But, one part of ∠D is 60o, so remaining part of ∠D = 110° – 60° = 50°
Now, considering the angles A and <D, their sum is equals to 180°
i.e., ∠A + ∠D = 180°
∠A = 180° – ∠D
= 180° – 110° = 70°
But, one part of ∠A is 30°, so remaining part of ∠A = 70° – 30° = 40°
It means, we get two angles of the triangle which are 50° and 40°
So, third angle of the triangle = 180° – (50° + 40°)
= 180° – (90°)
= 90°
Hence, all three angles of the triangles are 50°, 40°, 90°
Intext Questions and Answers
Question 1.
Find the unknown angle in the following figures?
Answer:
i. By extending the parallel lines and slanting line we get the figure below
From the figure, it is clear that two small angles are formed when the slanting line intersects the top and bottom parallel lines. Since these angles are equal in measure.
Therefore, the unknown angle is 45°
ii. By extending the parallel lines and slanting line we get the figure below
From the figure, it is clear that two small angles are formed when the slanting line intersects the top and bottom parallel lines. Since these angles are equal in measure.
Therefore, the unknown angle is 40°.
iii. By drawing a vertical line, as shown in the figure below, we can observe that this vertical line is parallel to the other parallel lines and bisects the middle angle. Therefore, the angles on the left side of the vertical line are 30°
Similarly, since the angle on the right side is also 30°, then the unknown angle will also be 30o.
When two parallel lines are cut by a third line (called a slanting line), different pairs of angles are formed.
Question 2.
One angle of a triangle is 72°. The other two angles are of equal measure. What are their measures?
Answer:
Sum of two other angles = 180° – 72° = 108°
Since the other two angles are equal, each angle is = \(\frac{108^0}{2}\) = 54
Question 3.
When a line crosses another line, how many angles are formed between them ?
Answer:
When a line crosses another line, then 4 angles are formed between them.
When two lines cross perpendicular, then all angles are 90°.
Consider the figure below,
Here, 4 angles are 21, 22, 23, 24, where 21 and 22 are small angles, and 23 and 24 are large angles Then we can say that,
The two small angles are of the same measure.
i.e., ∠1 = ∠2
The two large angles are of the same measure.
i.e., ∠3 = ∠4
The sum of a small angle and a large angle is 180°.
i.e., ∠1 + ∠4 = 180° and ∠3 + ∠4 = 180°.
Consider the figure below, which shows two parallel lines intersected by another line, forming eight angles.
Then we can say that,
∠1 = ∠5
∠2 = ∠6
∠3 = ∠7
∠4 = ∠8
For example,
In the given figure, one of the angles is given as 50°
Then, the remaining angles are shown below
A line intersects two parallel lines at angles of the same measure
Question 4.
Can you calculate the other seven angles which the parallel lines make with the slanting line?
Answer:
Class 7 Maths Chapter 1 Kerala Syllabus Parallel Lines Questions and Answers
Question 1.
Draw the parallelogram below with the given measures.
Answer:
Draw a horizontal line segment AB that is 6 cm long. This will be one side of the parallelogram.
At point A, use a protractor to measure and mark an angle of 50° from line AB.
From point A, draw a line segment AC that is 4 cm long, making sure it forms the 50° angle with AB.
From point B, draw a line segment BD that is 4 cm long, parallel to AC. You can use a ruler and a set square to ensure parallelism.
From point C, draw a line segment CD that is 6 cm long, parallel to AB. The points C and D should connect to complete the parallelogram.
Question 2.
The top and bottom lines in the figure are parallel. Find the unknown angle shown in the figure.
Answer:
Draw a horizontal line parallel to the other two parallel lines
From the figure, we get,
∠EAB = ∠ABD = 30°
∠DBC = ∠BCF = 60°
∴ ∠ABC = ∠ABD + ∠DBC
= 30° + 60° = 90°
Question 3.
Two vertical lines in the figure are parallel. Find the unknown angle shown in the figure.
Answer:
Consider the figure below
Given that AB and CD are parallel line
Draw a line PQ parallel to both AB and CD
∠A and ∠AQP are alternate angles
Therefore, ∠A = ∠AQP = 30°
Similarly,
∠C and ∠CQP are alternate angles
Therefore, ∠C = ∠CQP = 40°
But, ∠Q = ∠AQP + ∠CQP
= 30° + 40° = 70°
Question 4.
The figure shows a triangle drawn in a rectangle. Calculate the angles of the triangle.
Answer:
Consider the figure below,
Since, ABCD is rectangle ∠P, ∠Q, ∠R, ∠S equals to 90°
Now, it is given that one part of ∠Q is 50°, so the remaining part of∠A is 40°
Similarly, one part of ∠R is 30°, so the remaining part of ∠R is 60°
It implies that two angles of the triangle are 40° and 60o.
Therefore, the third angle of the triangle is = 180° – (40° + 60°)
= 180° — (100°) = 80°
So, the angles of triangles are 40°, 60°, 80°
Question 5.
A triangle is drawn inside a parallelogram. Calculate the angles of the triangle.
Answer:
Consider the figure below
∠D and ∠B are opposite angles of parallelogram.
Therefore, ∠D = ∠B = 105°
But, one part of ∠D is 50°, so remaining part of ∠D = 105° – 50° = 55°
Now, considering the angles ∠A and ∠D, their sum is equals to 180°
i.e., ∠A + ∠D = 180°
∠A = 180° – ∠D
= 180° – 105° = 75o
But, one part of ∠A is 20°, so remaining part of ∠A = 75° – 20o = 55°
It means, we get two angles of the triangle which are 55° and 55°
So, third angle of the triangle = 180° – (55° + 55°)
= 180° – (110°) = 70°
Hence, all three angles of the triangles are 55°, 55°, 70°.
Practice Questions
Question 1.
Draw the parallelogram below with the given measures.
Calculate the other three angles.
Answer:
The other three angles of Parallelogram 60°, 60°, 120°
Question 2.
Given that PQ and QR are parallel lines, find the angle shown in the figure?
Answer:
90°
Question 3.
In the figure, the pair of lines slanted to the left are parallel; and also the pair of lines slanted to the right.
Draw this figure:
Answer:
third angle of the bottom triangle = 90°
Question 4.
The top and bottom lines in the figure are parallel.
Calculate the third angle of the bottom triangle and all angles of the top triangle.
Answer:
all angles of the top triangle = 90°, 40o, 50°,
Question 5.
The left and right lines in the figure are parallel.
Calculate the third angle of the larger triangle and all angles of the smaller triangle.
Answer:
Third angle of the larger triangle = 80°
Question 6.
A triangle is drawn inside a parallelogram.
Calculate the angles of the triangle.
Answer:
All angles of smaller triangle = 80°, 55°, 45°
Question 7.
The figure shows a triangle drawn in a rectangle.
Calculate the angles of the triangle.
Answer:
All angles of triangle = 70°, 70°, 40°
Question 8.
Draw the triangle with the given measures.
Answer:
90°, 55°, 35° 70°.
Class 7 Maths Chapter 1 Notes Kerala Syllabus Parallel Lines
Understanding geometry begins with recognising the relationships between lines and angles. In this chapter, we delve into the fascinating world of parallel lines and the angles they create when intersected by other lines. This exploration will lay the foundation for more advanced geometric concepts and develop your ability to visualise and solve geometric problems.
Lines and Angles
We start by examining lines and the angles formed when they intersect. A special focus is given to pairs of angles that are formed on the same side of the slanting line, both above and below the parallel lines. These pairs of angles are equal in measure, helping us understand the special properties of angles created when parallel lines are intersected by a slanting line.
Matching Angles
Next, we explore the idea of matching angles. This section introduces alternate angles and co-interior angles more explicitly. Alternate angles are pairs of angles that lie on opposite sides of the transversal but inside the two lines. These angles are crucial in identifying and proving the parallel nature of lines. Co-interior angles are also revisited here, reinforcing their properties and significance.
Triangle Sum
Finally, we turn our attention to triangles, a fundamental shape in geometry. One of the most important properties of triangles is the sum of their interior angles. You will learn and prove that the total sum of all three angles in any triangle is always 180°. This section will include various exercises to solidify your understanding and application of this essential geometric principle.
By the end of this chapter, you will have a solid understanding of how parallel lines interact with transversals, the relationships between the resulting angles, and the fundamental properties of triangles. These concepts are not only crucial for your current studies but also form the bedrock of many advanced topics in geometry.
Matching Angles
Of the angles made when two parallel lines are cut by a slanting line,
-
- the small angles are of the same measure.
- the large angles are of the same measure.
- a small angle and a large angle add up to 180°
- If the intersecting line is perpendicular to one of the parallel lines, it would be perpendicular to the other line too, and all angles would be right angles.
Interior Angles:
These are the angles on the inside of the parallel lines.
When you look at the pairs of interior angles on the same side of the slanting line, they are called co-interior angles.
The sum of each pair of co-interior angles is always 180°.
Exterior Angles:
These are the angles on the outside of the parallel lines.
Similarly, when you look at the pairs of exterior angles on the same side of the slanting line, they are called co-exterior angles.
The sum of each pair of co-exterior angles is also 180°.
Alternate Interior Angles:
Alternate interior angles are pairs of angles that lie between the two parallel lines and on opposite sides of the slanting line.
These angles are same measure
Alternate Exterior Angles:
Alternate exterior angles are pairs of angles that lie outside the two parallel lines and on opposite sides of the slanting line.
Like alternate interior angles, alternate exterior angles are also equal.
Corresponding Angles:
These angles are in the same relative position at each intersection where a slanting line crosses the parallel lines.
Each pair of corresponding angles has the same measure.
Find out the other angles in the given parallelogram?
Answer:
First, consider the given angle (55° angle). To determine other angles, observe the intersections made by the left side with the top and bottom parallel lines.
The 55° angle and the angle above it form a pair of angles that add up to 180°. Therefore, the top angle is:
180° – 55° = 125°
Next, look at the angle to the right of the marked 55° angle.
To find this angle, examine the angles formed by the left and right parallel sides with the bottom line. The 55° angle and the angle to its right also form a pair of angles that add up to 180°.
Thus, this angle is also 125°, as calculated earlier.
Triangle Sum
The sum of all angles of a triangle is 180°
- A line intersects two parallel lines at angles of the same measure
- When two parallel lines are cut by a slanting line
- the small angles are of the same measure.
- the large angles are of the same measure.
- a small angle and a large angle add up to 180°.
- If the intersecting line is perpendicular to one of the parallel lines, would be perpendicular to the other line too, and all angles would be right angles.