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Parallel and Intersecting Lines - Parallel and Perpendicular Lines in Paper Folding

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Two lines are said to be parallel if they never meet, no matter how far they are extended. In paper folding, if you fold a paper into a rectangular strip and then unfold it, the vertical creases formed are parallel to each other. The perpendicular distance between these parallel creases remains constant throughout.

Parallel creases on a rectangular paper.
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Perpendicular lines are formed when two lines intersect at a right angle (90∘90^\circ). In paper folding, if you have a crease ll and you fold the paper so that one part of the crease ll falls exactly on top of the other part, the new crease mm formed will be perpendicular to ll.

Perpendicular creases forming a 90 degree angle.
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A key property discovered through folding is that if two different creases are both perpendicular to the same third crease, then those two creases must be parallel to each other. Symbolically, if l⊥nl \perp n and m⊥nm \perp n, then l∥ml \parallel m.

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Intersecting lines are lines that cross each other at a single point. In paper folding, any two folds that are not parallel and are not on the same line will eventually intersect at a point.

📐Formulae

l∥m  ⟺  Distance between l and m is constantl \parallel m \iff \text{Distance between } l \text{ and } m \text{ is constant}

l⊥m  ⟹  Angle of intersection=90∘l \perp m \implies \text{Angle of intersection} = 90^\circ

If l⊥n and m⊥n, then l∥m\text{If } l \perp n \text{ and } m \perp n, \text{ then } l \parallel m

💡Examples

Problem 1:

A student folds a rectangular sheet of paper twice. The first fold creates crease ll. The second fold is made by folding the paper such that crease ll falls exactly on itself, creating crease mm. What is the angle between crease ll and crease mm?

Solution:

The angle between crease ll and crease mm is 90∘90^\circ.

Explanation:

When a line (crease) is folded onto itself, the fold line formed is the perpendicular bisector of the overlapping parts. Therefore, the new crease mm is perpendicular to the original crease ll, meaning l⊥ml \perp m and the angle is 90∘90^\circ.

Problem 2:

Suppose you have a crease ABAB on a paper. You fold the paper to get a crease PQPQ such that PQ⊥ABPQ \perp AB. Then, you fold the paper again to get a crease XYXY such that XY⊥PQXY \perp PQ. Describe the relationship between ABAB and XYXY.

Solution:

AB∥XYAB \parallel XY

Explanation:

According to the properties of lines, if two different lines are perpendicular to the same transversal line, they must be parallel to each other. Here, both ABAB and XYXY are perpendicular to PQPQ. Therefore, ABAB is parallel to XYXY (AB∥XYAB \parallel XY).

Problem 3:

If the distance between two parallel creases ll and mm is 5 cm5\text{ cm} at one end of the paper, what will be the distance between them at the other end?

Solution:

5 cm5\text{ cm}

Explanation:

By definition, parallel lines (l∥ml \parallel m) maintain a constant perpendicular distance between them at all points. Therefore, the distance remains 5 cm5\text{ cm}.

Problem 4:

Draw a line ABAB on a paper. Fold the paper to create a crease CDCD that is perpendicular to ABAB. Now, create another crease EFEF by folding the paper such that EFEF is perpendicular to CDCD. What is the relationship between line ABAB and line EFEF?

Diagram showing lines AB and EF parallel to each other and both perpendicular to line CD.

Solution:

  1. Crease CD⊥ABCD \perp AB.
  2. Crease EF⊥CDEF \perp CD.
  3. Since both ABAB and EFEF are perpendicular to the same line CDCD, then AB∥EFAB \parallel EF.

Explanation:

When two lines are perpendicular to the same transversal line, the corresponding angles (which are 90∘90^\circ) are equal, making the lines parallel.

Problem 5:

A square sheet of paper is folded exactly in half along its diagonal to create crease L1L_1. Then it is unfolded and folded in half along the other diagonal to create crease L2L_2. At what angle do L1L_1 and L2L_2 intersect?

Square paper with two diagonal creases intersecting at the center.

Solution:

In a square, the diagonals bisect each other at right angles. Therefore, the angle of intersection between L1L_1 and L2L_2 is 90∘90^\circ.

Explanation:

Creases L1L_1 and L2L_2 are the diagonals of the square. By geometric property, the diagonals of a square are perpendicular to each other (L1⊥L2L_1 \perp L_2).