HOTS Question Answers: Parallel and Intersecting Lines | Mathematics (Ganita Prakash) Class 7 - New NCERT PDF Download

Q1: A student uses a set square and ruler to draw two lines that are both perpendicular to a base line. A transversal is then drawn such that it forms a 50° angle with the base line. What is the measure of the corresponding angles formed by the transversal with the two perpendicular lines? Explain why the two lines are parallel.

Ans: ​​​​

The two lines drawn are both perpendicular to the same base line, which means they each make a 90° angle with the base line.

Step 1: Understanding the setup

• Since both lines are perpendicular to the base line, they are parallel to each other (because lines perpendicular to the same line are parallel).

• A transversal is then drawn, forming a 50° angle with the base line.

Step 2: Finding corresponding angles

• The transversal makes a 50° angle with the base line, which means it forms an angle of 50° with one of the parallel lines.

• By the corresponding angles rule, the angle it forms with the second parallel line will also be 50°.

Step 3: Explaining why the lines are parallel

• Lines that are both perpendicular to the same line are parallel to each other.

• Since the transversal forms equal corresponding angles (50°) with both lines, it confirms that the lines are parallel.

Q2: A transversal intersects two parallel lines, creating an angle of 50°. What are the measures of the other seven angles formed? Explain your reasoning using the properties of parallel lines and transversals.

Ans: ​​​​

When a transversal intersects two parallel lines, it forms 8 angles in total — 4 at each intersection point.

Let’s assume the given angle of 50° is at the top right position of the first intersection.

Step 1: Use the property of Vertically Opposite Angles

• Vertically opposite angles are equal.

• So, the angle directly across from 50° is also 50°.

Step 2: Use the property of Linear Pair (Supplementary Angles)

• Angles on a straight line add up to 180°.

• So, the angle adjacent to 50° = 180° − 50° = 130°.

• Its vertically opposite angle is also 130°.

Now, we have found all 4 angles at the first intersection:

• Two angles of 50°

• Two angles of 130°

Step 3: Use the property of Corresponding Angles

• Corresponding angles at the second intersection (on the other parallel line) are equal to the ones at the first intersection.

• So, the angles at the second intersection will also be:

  • Two angles of 50°

  • Two angles of 130°

Q3: Two lines are intersected by a transversal, and one pair of interior angles on the same side of the transversal adds up to 180°. One of the angles is 65°. Find the measures of the corresponding angles to these two angles, and prove the lines are parallel.

Ans:
When two lines are intersected by a transversal, and the interior angles on the same side of the transversal add up to 180°, it is a property that shows the lines are parallel.

Step 1: Find the other angle in the same-side interior pair.
Given one angle = 65°
Sum of same-side interior angles = 180°
Other angle = 180° − 65° = 115°

Step 2: Find the corresponding angles.

• The angle corresponding to 65° is also 65° (by corresponding angles rule).

• The angle corresponding to 115° is also 115°.

Step 3: Prove the lines are parallel.

• Since the same-side interior angles add up to 180°, and

• The corresponding angles are equal,
  we conclude that the two lines are parallel.

Q4: A transversal intersects two lines, forming a pair of alternate interior angles where one angle is 110°. The other pair of alternate interior angles includes an angle of 70°. Are the two lines parallel? If not, what is the difference between the angles in the second pair of alternate interior angles?

Ans: For two lines to be parallel, the alternate interior angles formed by a transversal must be equal.

• In the first pair of alternate interior angles, one angle is 110°.

• This means the angle on the opposite side should also be 110° if the lines are parallel.

• In the second pair, one of the alternate interior angles is given as 70°.

Since 110° ≠ 70°, the alternate interior angles are not equal, which means the two lines are not parallel.

To find the difference between the angles in the second pair:

• One angle = 70°, and if the lines were parallel, the other angle would also be 70°.

• But here, since the lines are not parallel and one angle is 70°, the other angle in the pair must be 110° (as given in the opposite pair).

Difference = 110° − 70° = 40°.

Therefore, the lines are not parallel, and the difference between the angles in the second pair of alternate interior angles is 40°.

Q5: In an optical illusion, a student draws 4 horizontal parallel lines on a sheet of paper. Between each pair of lines, they draw zigzag patterns that make the lines appear slanted. If a transversal is drawn across these lines forming a corresponding angle of 120° with the first line, what are the corresponding angles with the other three lines? Why does the illusion make the lines appear non-parallel?

Ans: 

Since the four lines are parallel and a transversal crosses all of them, the corresponding angles formed with each line will be the same.

• The angle formed with the first line is 120°.

• Therefore, the corresponding angles with the other three lines are also 120° each.

This is because corresponding angles are equal when a transversal crosses parallel lines.

The illusion makes the lines appear non-parallel because of the zigzag patterns drawn between the lines. These patterns trick the eyes and brain by creating a false sense of direction and tilt, which causes the parallel lines to appear slanted or not equally spaced. This is a common example of a visual illusion where our perception does not match the actual geometry.