NCERT Solutions Class 7 Maths Chapter 5 Parallel and Intersecting Lines

The Class 7 Maths Chapter 5 Parallel and Intersecting Lines introduces students to the basic concepts of geometry. In this chapter, students learn how to identify lines that never meet (parallel lines) and lines that cross each other (intersecting lines). With the help of diagrams and real-life examples like railway tracks or road crossings, the chapter makes it easy to understand how these lines work and where we see them around us.

These Class 7 NCERT Solutions for the chapter Parallel and Intersecting Lines follow the latest NCERT syllabus and explain all its textbook questions in a clear and step-by-step way. They help students understand important terms like transversal, angles formed by lines, and how to recognize parallel lines using properties of angles.

1.0NCERT Solutions Class 7 Maths Chapter 5 Parallel and Intersecting Lines – Download PDF

Get ALLEN’s exclusive and simplified NCERT Solutions Class 7 Maths PDF for Chapter 5 – Parallel and Intersecting Lines. Our expert-prepared answers are aligned with the NCERT syllabus and perfect for concept clarity, exam prep, and revision.

2.0Key Concepts in Chapter 5: Parallel and Intersecting Lines

1. What is a Line?

• A line is a straight path that extends infinitely in both directions.
• It has no endpoints and is usually denoted by small letters like l, m, n.

2. Line Segments and Rays

• A line segment has two endpoints. It is a measurable part of a line.
• A ray has one endpoint and extends infinitely in one direction.
• These form the building blocks of geometry.

3. Intersecting Lines

• Two lines that meet or cross each other at a point are called intersecting lines.
• The point at which they meet is called the point of intersection.
• Example: The diagonals of a square are intersecting lines.

4. Parallel Lines

• Lines that never meet, no matter how far they are extended, are called parallel lines.
• The distance between parallel lines is always equal.
• Example: Railway tracks or opposite edges of a notebook.

5. Perpendicular Lines

• A special case of intersecting lines is perpendicular lines, which intersect at a 90-degree angle.
• Represented using the ⊥ symbol.

6. Examples from Daily Life

• Parallel lines: Zebra crossings, steps of a ladder, railway tracks.
• Intersecting lines: Scissors, road junctions, corners of a room.

7. Drawing Lines Using a Ruler and Set Square

• Students are guided on how to draw precise parallel and intersecting lines using a scale, set square, and protractor.
• This also introduces basic construction skills used in geometry.

3.0NCERT Solutions for Chapter 5 Parallel and Intersecting Lines : All Exercises

Exercise 5.1

The Exercise 5.1 introduces students to the basic relationships between angles formed when two lines intersect. Learners identify linear pairs and vertically opposite angles in geometric diagrams and understand how these angle pairs are related. The exercise strengthens the foundation of angle relationships that are important for further geometry concepts.

Key Concepts Covered:

• Linear pair of angles
• Vertically opposite angles
• Angle relationships in intersecting lines
• Identifying angle pairs in diagrams

Exercise 5.2

Exercise 5.4 of chapter 5 focuses on finding unknown angles formed when parallel lines are intersected by a transversal. Students apply properties of corresponding angles, alternate interior angles, and co-interior angles to determine missing angle measures in diagrams. This exercise develops reasoning skills while working with geometric relationships.

Key Concepts Covered:

• Alternate interior angles
• Corresponding angles
• Co-interior angles
• Finding unknown angles using angle properties

Exercise 5.3

This exercise involves solving geometry problems using multiple angle relationships in diagrams with parallel lines and transversals. Students analyse intersecting lines and apply angle properties step by step to determine missing angles. The exercise strengthens logical reasoning and diagram interpretation skills.

Key Concepts Covered:

• Understadn angle relationships in complex figures
• learn about Parallel lines with transversals
• Multi-step angle calculations
• Interpreting geometric diagrams

Exercise 5.4

The exercise 5.8 extends angle reasoning by applying parallel line properties in combined geometric situations. Students solve problems involving several intersecting lines and multiple angle relationships, using known angle values to calculate unknown ones accurately. This exercise improves analytical thinking in geometry.

Key Concepts Covered:

• Combined angle relationships
• Reasoning with parallel lines
• Advanced angle calculations
• Geometry problem solving

4.0NCERT Class 7 Maths Chapter 5 Parallel and Intersecting Lines: Detailed Solutions

FIGURE IT OUT-01

• List all the linear pairs and vertically opposite angles you observe in.
  

Sol.

FIGURE IT OUT-02

• Draw some lines perpendicular to the lines given on the dot paper in Fig.
  
  Sol.
  
• In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol. (a) How did you spot the perpendicular lines? (b) How did you spot the parallel lines?
  
  Sol.
  
  (a) To spot perpendicular lines in a geometric figure, observe if lines intersect at $90^{\circ }$. (b) To spot parallel lines in a geometric figure, observe if lines never intersect at any point.
• In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints. Sol. Let's draw some lines on a dot paper
  
• Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.
  
  (a) Did you find it challenging to draw some of them? (b) Which ones? (c) How did you do it? Sol.
  
  (a) Yes, it can be more challenging to intuitively draw lines parallel to some of the segments compared to others. (b) The most challenging lines are typically those with less common slopes, like $g$ and $h$, because it's harder to judge their angle by eye compared to simple horizontal, vertical, or 45-degee lines. (c) The best ways to draw a parallel line on the dot paper is to count the dot pattern (the "slope"). For example, line c goes 'up 1 dot and right 2 dots'. To draw a parallel line, you simply start at a new dot and repeat that same 'up 1, right 2' movement.
• In Figure which line is parallel to line a , line $b$ or line $c$ ? How do you decide this?
  
  Sol. In the given figure, "Line a is parallel to line c because they lie in the same plane and do not intersect, regardless of how far they are extended." We notice that by extending both lines and noticing that they never meet.

FIGURE IT OUT-03

• Can you draw a line parallel to l , that goes through point A? How will you do it with the tools from your geometry box? Describe your method. A
  
  Sol. Tools needed : Ruler, Set-squares (right-angled triangle), Pencil
  
  
  Steps: 1.Place the set square so that one side is along the line l. 2.Hold the ruler against the other side of the set square (the ruler won't move). 3.Slide the set square along the ruler until one side reaches point A . 4.Draw a line along the edge of the set square through point A. 5.This new line is parallel to line 1 and passes through point A. 6.Find the angles marked below.

FIGURE IT OUT-04

Find the angles marked below:

• Find the angle represented by a.
  
  
  
  
  Sol.
  
  Here, $\angle 1$ is $42^{\circ }$, so $\angle 2=180^{\circ }-42^{\circ }=138^{\circ }$, ( $\angle 1$ and $\angle 2$ form a linear pair) Now, since lines l and m are parallel and t is a transversal. Therefore, $\angle 2=\mathrm{a}$ (alternate angles are qual) Thus, $\mathrm{a}=138^{\circ }$
  
  Here, $\angle 1=62^{\circ }$ So, $\angle 1+\angle 2=180^{\circ }$ ( $\angle 1$ and $\angle 2$ form a linear pair) $\angle 2=180^{\circ }-62^{\circ }=118^{\circ }$ Now, $\angle 2=\angle 3$ (corresponding angles are equal) So, $\angle 3=118^{\circ }$ Now, $\angle 3=\mathrm{a}$ (corresponding angles are equal) Thus, $\mathrm{a}=118^{\circ }$
  
  Here, lines $s$ and $l$ are intersecting lines. So, $\angle 1=110^{\circ }$ [Vertically opposite angles] And $\angle 1=\angle 2=110^{\circ }$ because lines $\ell$ and m are parallel and line s is a transversal. Therefore $\angle 3=\angle 2-35^{\circ }=110^{\circ }-35^{\circ }=75^{\circ }$ Also, $\angle 3=\angle 4=75^{\circ }$ [Corresponding angles] So, ${\mathrm{a}}^{\circ }=180^{\circ }-75^{\circ }=105^{\circ }$ [Linear pair angles]
  
  Using angles on a straight line, we have $\angle 1+\angle 2+67^{\circ }=180^{\circ }$ $\angle 2=180^{\circ }-67^{\circ }-90^{\circ }$ $\angle 2=23^{\circ }$ [Since $\angle 1=90^{\circ }$ ] Thus, $\mathrm{a}=23^{\circ }$ as $\angle 2=\angle \mathrm{a}$ [Alternate angles as lines $l$ and $t$ are parallel]
• In the figures below, what angles do x and y stand for ?
  
  Sol.
  
  Since lines $s$ and $m$ are perpendicular to each other. So $\angle 2=90^{\circ }$ Now, $\angle 2+65^{\circ }+{\mathrm{x}}^{\circ }=180^{\circ }$ [Linear Pair] So, $x^{\circ }=180^{\circ }-90^{\circ }-65^{\circ }=25^{\circ }$ Now, lines t and m are two intersecting lines. So, $\mathrm{x}=\angle 1=25^{\circ }$ [Vertically Opposite Angles] Lines l and m are parallel to each other, and t is a transversal. So, ${\mathrm{y}}^{\circ }=\angle 2+65^{\circ }=90^{\circ }+65^{\circ }=155^{\circ }$ [Alternate angles] Therefore, $\mathrm{y}=155^{\circ }$.
  
  Since lines l and m are parallel and line s is a transversal. So, $\angle 3=78^{\circ }$ [Alternate Angles] Also, lines l and m are parallel, and line t is a transversal. So, $\angle 1=53^{\circ }$ [Alternate Angles] Therefore, $\angle 2=\angle 3-\angle 1=78^{\circ }-53^{\circ }=25^{\circ }$ Lines s and t are intersecting lines. Therefore, ${\mathrm{x}}^{\circ }=\angle 2=25^{\circ }$ [Vertically Opposite Angles]
• In Fig. $\angle \mathrm{ABC}=45^{\circ }$ and $\angle \mathrm{IKJ}=78^{\circ }$. Find angles $\angle \mathrm{GEH},\angle \mathrm{HEF},\angle \mathrm{FED}$.
  
  Sol. Line segments IA and HC intersect at point $B$. So, $\angle \mathrm{ABC}=\angle \mathrm{KBE}=45^{\circ }$ [Vertically Opposite Angles] Similarly, line segments JF and IA intersect at point K . So, $\angle \mathrm{IKJ}=\angle \mathrm{BKE}=78^{\circ }$ [Vertically Opposite Angles] $\angle \mathrm{KBE}=\angle \mathrm{GEH}=45^{\circ }$ [Corresponding Angles] Similarly, $\angle \mathrm{BKE}=\angle \mathrm{FED}=78^{\circ }$ [Corresponding Angles] Now, $\angle \mathrm{GEH}+\angle \mathrm{HEF}+\angle \mathrm{FED}=180^{\circ }$ [Linear Pair] $\angle \mathrm{HEF}=180^{\circ }-45^{\circ }-78^{\circ }=57^{\circ }$
• In Fig. AB is parallel to CD and CD is parallel to EF . Also, EA is perpendicular to AB . If $\angle \mathrm{BEF}=55^{\circ }$, find the values of x and y.
  
  Sol. Given AB is parallel to CD and CD is parallel to EF . So, AB is parallel to EF . Now, EF is parallel to CD , and DE is a transversal. So, ${\mathrm{y}}^{\circ }+55^{\circ }=180^{\circ }$ [Sum of interior angles] $\mathrm{y}=125^{\circ }$ Now, AB is parallel to CD , and BD is a transversal. So, ${\mathrm{x}}^{\circ }={\mathrm{y}}^{\circ }=125^{\circ }$ [Corresponding Angles]
• What is the measure of angle $\angle \mathrm{NOP}$ in Fig. given that $\mathrm{LM}\Vert \mathrm{PQ}$.
  
  Sol. Draw a line RS through N, which is parallel to line LM , and line TU through 0 , which is parallel to line PQ .
  
  $\angle \mathrm{LMN}=\angle \mathrm{MNS}$ [Alternate Angles] Therefore, ${\mathrm{w}}^{\circ }=40^{\circ }$ Given, $\angle \mathrm{MNO}=96^{\circ }$ ${\mathrm{w}}^{\circ }+{\mathrm{x}}^{\circ }=96^{\circ }$ $x^{\circ }=96^{\circ }-40^{\circ }=56^{\circ }$ Now, RS is parallel to TU, and NO is a transversal. So, $\angle \mathrm{SNO}=\angle \mathrm{NOT}$ [Alternate Angles] Therefore, ${\mathrm{y}}^{\circ }={\mathrm{x}}^{\circ }=56^{\circ }$ Now TU is parallel to PQ , and OP is a transversal. So, $\angle \mathrm{TOP}=\angle \mathrm{OPQ}$ [Alternate Angles] ${\mathrm{z}}^{\circ }=52^{\circ }$ [Given $\angle \mathrm{OPQ}=52^{\circ }$ ] Thus, ${\mathrm{a}}^{\circ }={\mathrm{y}}^{\circ }+{\mathrm{z}}^{\circ }=56^{\circ }+52^{\circ }=108^{\circ }$

5.0Key Features of NCERT Solutions Class 7 Maths Chapter 5 : Parallel and Intersecting Lines

• Detailed Concept Explanations: Each solution is designed to reinforce the understanding of key geometric concepts, from basic definitions of lines and angles to the intricate relationships formed by transversals intersecting parallel lines.
• Step-by-Step Problem Solving: Solutions to all exercises from the NCERT Ganita Prakash textbook are presented with clear, easy-to-follow steps. This helps students grasp the logic behind each solution and develop effective problem-solving skills.
• Diagrammatic Representations: Geometry relies heavily on visual understanding. Our solutions frequently include diagrams to illustrate concepts like vertically opposite angles, corresponding angles, and alternate angles, making complex ideas more accessible.
• Real-World Applications: The solutions connect theoretical concepts to practical examples found in daily life, such as the angles formed by roads or architectural designs, highlighting the relevance of geometry.
• Adherence to NCERT Syllabus: Our solutions are strictly aligned with the updated NCERT Ganita Prakash curriculum for Class 7, ensuring that students study the most current and relevant content for their examinations.
• Focus on Properties and Theorems: Emphasis is placed on understanding and applying the properties of angles formed by intersecting and parallel lines, which are critical for solving geometric problems.
• Downloadable PDF: Conveniently available in a PDF format, these NCERT Solutions allow for easy access and offline study, enabling flexible and effective learning for all students.