Lines and Angles – CBSE Class 7 Maths

Introduction to Lines and Angles

Lines and Angles In mathematics, geometry is a branch that deals with shapes, sizes, and the properties of space. Chapter 5, “Lines and Angles,” introduces the fundamental building blocks of geometry: lines and angles. These concepts are essential for understanding shapes, patterns, and measurements in both two-dimensional and three-dimensional spaces. This chapter explores different types of lines, angles, and their relationships, along with properties like complementary angles, supplementary angles, and the behavior of angles formed by a transversal intersecting parallel lines.

---

1. Basic Terms and Definitions

• Point: A point is a precise location in space with no length, width, or height. It is represented by a dot and named with a capital letter (e.g., Point A).
• Line: A line is a straight, one-dimensional figure that extends infinitely in both directions. It is represented by a straight line with arrows at both ends (e.g., AB↔ \overleftrightarrow{AB} AB).
• Line Segment: A line segment is a part of a line with two endpoints. It has a definite length and is denoted as AB‾ \overline{AB} AB.
• Ray: A ray is a part of a line that starts at one endpoint and extends infinitely in one direction. It is denoted as AB→ \overrightarrow{AB} AB, where A is the starting point.
• Angle: An angle is formed when two rays share a common endpoint. The common endpoint is called the vertex, and the rays are called the arms of the angle. Angles are denoted using the symbol $\angle$ (e.g., $\angle ABC$).

---

2. Types of Angles :Lines and Angles

Angles are classified based on their measure in degrees (°). A protractor is used to measure angles.

• Acute Angle: An angle that measures less than 90° (e.g., 45°).
• Right Angle: An angle that measures exactly 90°. It is often represented by a small square at the vertex.
• Obtuse Angle: An angle that measures more than 90° but less than 180° (e.g., 120°).
• Straight Angle: An angle that measures exactly 180°. It looks like a straight line.
• Reflex Angle: An angle that measures more than 180° but less than 360° (e.g., 250°).
• Complete Angle: An angle that measures exactly 360°. It represents a full rotation around a point.

---

3. Related Angles

Angles often have relationships with each other based on their positions or measures. These relationships are key to solving problems in geometry.

a) Complementary Angles

• Two angles are complementary if the sum of their measures is 90°.
• Example: If ∠A=30∘ \angle A = 30^\circ ∠A=30∘, then its complementary angle is 90∘−30∘=60∘ 90^\circ – 30^\circ = 60^\circ 90∘−30∘=60∘.
• Complementary angles may or may not be adjacent (sharing a common arm and vertex).

b) Supplementary Angles

• Two angles are supplementary if the sum of their measures is 180°.
• Example: If ∠P=110∘ \angle P = 110^\circ ∠P=110∘, then its supplementary angle is 180∘−110∘=70∘ 180^\circ – 110^\circ = 70^\circ 180∘−110∘=70∘.
• Like complementary angles, supplementary angles can be adjacent or non-adjacent.

c) Adjacent Angles

• Two angles are adjacent if they share a common vertex, a common arm, and do not overlap.
• Example: In a straight line, if ∠ABC=40∘ \angle ABC = 40^\circ ∠ABC=40∘ and ∠CBD=140∘ \angle CBD = 140^\circ ∠CBD=140∘ share vertex B and arm BC, they are adjacent and supplementary (since 40∘+140∘=180∘ 40^\circ + 140^\circ = 180^\circ 40∘+140∘=180∘).

d) Vertically Opposite Angles

• When two lines intersect, they form four angles. The angles opposite each other at the intersection point are called vertically opposite angles.
• Property: Vertically opposite angles are equal.
• Example: If two lines AB and CD intersect at point O, and ∠AOC=50∘ \angle AOC = 50^\circ ∠AOC=50∘, then the vertically opposite angle ∠BOD=50∘ \angle BOD = 50^\circ ∠BOD=50∘.

---

4. Pairs of Lines: Lines and Angles

Lines can interact in different ways, leading to specific angle relationships.

a) Intersecting Lines

• Two lines that cross each other at a single point are called intersecting lines.
• Example: Two roads meeting at a junction.
• At the point of intersection, vertically opposite angles are formed.

b) Parallel Lines

• Two lines that never intersect, no matter how far they are extended, are called parallel lines. They are always equidistant from each other.
• Parallel lines are denoted using the symbol $\Vert$ (e.g., $AB\Vert CD$).
• Example: Railway tracks.

c) Transversal

• A transversal is a line that intersects two or more lines at distinct points.
• When a transversal intersects two parallel lines, it creates several pairs of angles with special properties.

---

5. Angles Formed by a Transversal:Lines and Angles

When a transversal intersects two parallel lines, eight angles are formed. These angles have specific names and relationships:

• Corresponding Angles: Angles that are on the same side of the transversal and in matching positions relative to the parallel lines. They are equal.
  • Example: If $\angle 1$ and $\angle 5$ are corresponding angles, then $\angle 1=\angle 5$.
• Alternate Interior Angles: Angles that are on opposite sides of the transversal and inside the parallel lines. They are equal.
  • Example: If $\angle 3$ and $\angle 6$ are alternate interior angles, then $\angle 3=\angle 6$.
• Alternate Exterior Angles: Angles that are on opposite sides of the transversal and outside the parallel lines. They are equal.
  • Example: If $\angle 1$ and $\angle 8$ are alternate exterior angles, then $\angle 1=\angle 8$.
• Interior Angles on the Same Side of the Transversal: Angles that are on the same side of the transversal and inside the parallel lines. They are supplementary (sum = 180°).
  • Example: If $\angle 3$ and $\angle 5$ are interior angles on the same side, then ∠3+∠5=180∘ \angle 3 + \angle 5 = 180^\circ ∠3+∠5=180∘.

---

6. Properties of Parallel Lines and Transversal

• Corresponding Angles Property: If a transversal intersects two parallel lines, the corresponding angles are equal.
• Alternate Angles Property: If a transversal intersects two parallel lines, the alternate interior angles and alternate exterior angles are equal.
• Interior Angles Property: If a transversal intersects two parallel lines, the interior angles on the same side of the transversal are supplementary.
• Converse: If two lines are intersected by a transversal and any one of the above properties holds (e.g., corresponding angles are equal), then the lines are parallel.

---

7. Examples:Lines and Angles

1. Complementary Angles: If ∠XYZ=25∘ \angle XYZ = 25^\circ ∠XYZ=25∘, find its complementary angle.
   • Solution: 90∘−25∘=65∘ 90^\circ – 25^\circ = 65^\circ 90∘−25∘=65∘.
2. Supplementary Angles: If ∠PQR=135∘ \angle PQR = 135^\circ ∠PQR=135∘, find its supplementary angle.
   • Solution: 180∘−135∘=45∘ 180^\circ – 135^\circ = 45^\circ 180∘−135∘=45∘.
3. Vertically Opposite Angles: Two lines intersect, and one angle is 70°. Find the other angles.
   • Solution: Vertically opposite angle = 70°. Adjacent angles are supplementary, so 180∘−70∘=110∘ 180^\circ – 70^\circ = 110^\circ 180∘−70∘=110∘. Thus, angles are 70°, 70°, 110°, 110°.
4. Parallel Lines: A transversal intersects two parallel lines, and one angle is 50°. Find all angles.
   • Solution: Corresponding angles = 50°. Alternate interior angles = 50°. Interior angles on the same side = 180∘−50∘=130∘ 180^\circ – 50^\circ = 130^\circ 180∘−50∘=130∘. All eight angles: 50°, 130°, 50°, 130°, 50°, 130°, 50°, 130°.

---

8. Key Points to Remember:Lines and Angles

• Angles are measured in degrees using a protractor.
• Complementary angles sum to 90°, supplementary angles sum to 180°.
• Vertically opposite angles are always equal.
• Parallel lines with a transversal create equal corresponding and alternate angles, and supplementary interior angles on the same side.
• Practice drawing lines and angles to understand their relationships better.

Download pdf notes and Sulutions of the Chapter:

Chapter 6: The Triangle and its Properties

Please Visit Readspot for Hindi Medium Study Material