Class 7 Mathematics Notes Chapter 5 (Lines and Angles) – Mathematics Book
Alright class, let's get straight into Chapter 5: Lines and Angles. This is a fundamental chapter, and concepts learned here are crucial not just for your Class 7 exams but form the building blocks for geometry in higher classes and competitive exams. Pay close attention!
Chapter 5: Lines and Angles - Detailed Notes for Exam Preparation
1. Basic Definitions:
- Point: A location in space, represented by a dot. It has no size (length, width, or thickness). Named by a capital letter (e.g., Point A).
- Line Segment: The shortest distance between two points. It has two endpoints and a definite length. Denoted as AB (with a bar on top).
- Line: A line segment extended infinitely in both directions. It has no endpoints and no definite length. Denoted as AB (with arrows on both ends) or by a small letter (e.g., line l).
- Ray: A part of a line that starts at a point (called the endpoint or initial point) and extends infinitely in one direction. Denoted as OA (with an arrow on one end, starting from O).
- Angle: Formed when two rays originate from the same endpoint (called the vertex). The rays forming the angle are called its arms or sides. Measured in degrees (°). Named using three points (vertex in the middle, e.g., ∠ABC) or just the vertex (e.g., ∠B) if there's no confusion, or by a number/letter inside (e.g., ∠1).
2. Types of Angles (Based on Measure):
- Acute Angle: An angle whose measure is greater than 0° but less than 90°. (0° < angle < 90°)
- Right Angle: An angle whose measure is exactly 90°.
- Obtuse Angle: An angle whose measure is greater than 90° but less than 180°. (90° < angle < 180°)
- Straight Angle: An angle whose measure is exactly 180°. It forms a straight line.
- Reflex Angle: An angle whose measure is greater than 180° but less than 360°. (180° < angle < 360°)
3. Related Angles (Pairs of Angles):
- Complementary Angles: Two angles whose measures add up to 90°.
- Each angle is called the complement of the other.
- Example: 30° and 60° are complementary angles. The complement of 40° is (90° - 40°) = 50°.
- Supplementary Angles: Two angles whose measures add up to 180°.
- Each angle is called the supplement of the other.
- Example: 110° and 70° are supplementary angles. The supplement of 80° is (180° - 80°) = 100°.
- Adjacent Angles: Two angles that have:
- A common vertex.
- A common arm.
- Their non-common arms are on opposite sides of the common arm.
- Important Note: Adjacent angles don't necessarily add up to 90° or 180°, unless specified otherwise.
- Linear Pair: A pair of adjacent angles whose non-common sides form a straight line.
- Angles in a linear pair are always supplementary (add up to 180°).
- Think: 'Linear' -> Line -> Straight Angle -> 180°.
- Vertically Opposite Angles: Angles formed by two intersecting lines. They are opposite to each other at the vertex.
- Key Property: Vertically opposite angles are always equal.
- If two lines intersect, they form two pairs of vertically opposite angles.
4. Pairs of Lines:
- Intersecting Lines: Two lines that cross each other at a single point (the point of intersection).
- Parallel Lines: Two lines in a plane that never intersect, no matter how far they are extended. The distance between them is always constant. Denoted as l || m.
- Transversal: A line that intersects two or more distinct lines at distinct points.
5. Angles Made by a Transversal:
When a transversal intersects two lines (let's call them l and m), it forms eight angles. These angles have special names based on their positions:
- Interior Angles: Angles lying between the two lines l and m. (e.g., ∠3, ∠4, ∠5, ∠6 in standard diagrams)
- Exterior Angles: Angles lying outside the two lines l and m. (e.g., ∠1, ∠2, ∠7, ∠8 in standard diagrams)
Specific Pairs (Crucial for Parallel Lines):
- Corresponding Angles: Angles in the same relative position at each intersection.
- Pairs: (∠1, ∠5), (∠2, ∠6), (∠3, ∠7), (∠4, ∠8)
- Think: 'Top-left' at both intersections, 'Top-right' at both, etc.
- Alternate Interior Angles: Angles on opposite sides of the transversal and between the two lines l and m.
- Pairs: (∠3, ∠5), (∠4, ∠6)
- Think: 'Across' the transversal, 'Inside' the lines.
- Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the two lines l and m.
- Pairs: (∠1, ∠7), (∠2, ∠8)
- Think: 'Across' the transversal, 'Outside' the lines.
- Interior Angles on the Same Side of the Transversal (Consecutive Interior Angles or Co-interior Angles): Angles on the same side of the transversal and between the two lines l and m.
- Pairs: (∠3, ∠6), (∠4, ∠5)
6. Transversal of Parallel Lines (VERY IMPORTANT):
If a transversal intersects two parallel lines (l || m):
- Corresponding Angles are equal. (∠1=∠5, ∠2=∠6, ∠3=∠7, ∠4=∠8)
- Alternate Interior Angles are equal. (∠3=∠5, ∠4=∠6)
- Alternate Exterior Angles are equal. (∠1=∠7, ∠2=∠8)
- Interior Angles on the same side of the transversal are supplementary (add up to 180°). (∠3+∠6=180°, ∠4+∠5=180°)
7. Checking for Parallel Lines (Converse Properties):
If a transversal intersects two lines such that any one of the following conditions is true, then the two lines are parallel:
- If a pair of corresponding angles is equal.
- If a pair of alternate interior angles is equal.
- If a pair of alternate exterior angles is equal.
- If a pair of interior angles on the same side of the transversal is supplementary.
Key Takeaways for Exams:
- Know the definitions precisely.
- Memorize the conditions for complementary (90°) and supplementary (180°) angles.
- Understand the properties of linear pairs (supplementary) and vertically opposite angles (equal).
- Master the names and properties of angles formed by a transversal, especially when the lines are parallel.
- Be able to use the converse properties to determine if lines are parallel.
Multiple Choice Questions (MCQs):
-
The complement of an angle measuring 37° is:
A) 143°
B) 53°
C) 63°
D) 13° -
Two angles form a linear pair. If one angle is 80°, the other angle is:
A) 10°
B) 100°
C) 90°
D) 280° -
Two lines AB and CD intersect at point O. If ∠AOC = 50°, what is the measure of ∠BOD?
A) 40°
B) 50°
C) 130°
D) 90° -
In the context of a transversal intersecting two lines, angles lying between the two lines are called:
A) Exterior angles
B) Corresponding angles
C) Interior angles
D) Alternate exterior angles -
If two parallel lines are intersected by a transversal, then the alternate interior angles are:
A) Complementary
B) Supplementary
C) Equal
D) Unequal -
Find the angle which is equal to its supplement.
A) 45°
B) 90°
C) 180°
D) 60° -
A transversal intersects two lines l and m. If a pair of corresponding angles is equal (say 75° each), then the lines l and m are:
A) Intersecting
B) Perpendicular
C) Parallel
D) Cannot be determined -
In a pair of adjacent angles, (i) vertex is common, (ii) common arm, (iii) non-common arms are on opposite sides of the common arm. Which statements are correct?
A) Only (i)
B) Only (i) and (ii)
C) Only (ii) and (iii)
D) All (i), (ii), and (iii) -
Two parallel lines are cut by a transversal. If one interior angle on the same side of the transversal is 110°, what is the measure of the other interior angle on the same side?
A) 110°
B) 90°
C) 70°
D) 20° -
An angle is greater than 90° but less than 180°. This angle is called:
A) Acute angle
B) Right angle
C) Straight angle
D) Obtuse angle
Answer Key for MCQs:
- B (90° - 37° = 53°)
- B (180° - 80° = 100°)
- B (Vertically opposite angles are equal)
- C (Definition)
- C (Property of parallel lines)
- B (Let angle be x. x = 180° - x => 2x = 180° => x = 90°)
- C (Converse property: If corresponding angles are equal, lines are parallel)
- D (Definition of adjacent angles)
- C (Interior angles on the same side are supplementary for parallel lines: 180° - 110° = 70°)
- D (Definition)
Revise these notes thoroughly. Practice identifying these angles in different diagrams. Good luck with your preparation!