Quick Revision Notes: Class 7 Mathematics

Chapter: Lines and Angles

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1. Basic Terms and Definitions

• Point (बिंदु): A point determines a location. It has no length, width, or thickness. Represented by a capital letter (e.g., Point $A$).
• Line (रेखा): A line is a straight path that extends infinitely in both directions. It has no endpoints and no definite length. (Represented as $\overleftrightarrow{AB}$)
• Line Segment (रेखाखंड): A line segment is a part of a line that has two fixed endpoints. It has a definite length. (Represented as $\overline{AB}$)
• Ray (किरण): A ray is a part of a line that has one endpoint and extends infinitely in one direction. (Represented as $\overrightarrow{AB}$)
• Angle (कोण): An angle is formed when two rays originate from the same starting point. The common point is called the vertex (शीर्ष) and the two rays are the arms (भुजाएँ) of the angle.

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2. Types of Angles

• Acute Angle (न्यून कोण): An angle measuring less than $90^\circ$ ($0^\circ < \theta < 90^\circ$).
• Right Angle (समकोण): An angle measuring exactly $90^\circ$.
• Obtuse Angle (अधिक कोण): An angle measuring more than $90^\circ$ and less than $180^\circ$ ($90^\circ < \theta < 180^\circ$).
• Straight Angle (ऋजु कोण / सरल कोण): An angle measuring exactly $180^\circ$.
• Reflect Angle (प्रतिवर्ती कोण): An angle measuring more than $180^\circ$ and less than $360^\circ$ ($180^\circ < \theta < 360^\circ$).
• Complete Angle (पूर्ण कोण): An angle measuring exactly $360^\circ$.

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3. Pairs of Angles

(a) Complementary Angles (कोटिपूरक कोण / पूरक कोण)

• Definition: Two angles whose sum is equal to $90^\circ$.
• Formula: If $\angle 1$ and $\angle 2$ are complementary, then $\angle 1 + \angle 2 = 90^\circ$.
• Note: Each angle is the complement of the other.

(b) Supplementary Angles (संपूरक कोण)

• Definition: Two angles whose sum is equal to $180^\circ$.
• Formula: If $\angle 1$ and $\angle 2$ are supplementary, then $\angle 1 + \angle 2 = 180^\circ$.
• Note: Each angle is the supplement of the other.

(c) Adjacent Angles (आसन्न कोण)

• Properties:
  1. They have a common vertex.
  2. They have a common arm.
  3. Their non-common arms are on opposite sides of the common arm.

(d) Linear Pair (रैखिक युग्म)

• Definition: A linear pair is a pair of adjacent angles whose sum is $180^\circ$ (they form a straight line).
• Formula: $\angle 1 + \angle 2 = 180^\circ$ (when they lie on a straight line).

(e) Vertically Opposite Angles (शीर्षाभिमुख कोण)

• Definition: When two lines intersect, the angles formed at the intersection point opposite to each other are called vertically opposite angles.
• Key Rule: Vertically opposite angles are always equal.
• Formula: If lines $AB$ and $CD$ intersect at $O$, then:
  • $\angle 1 = \angle 3$
  • $\angle 2 = \angle 4$

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4. Pairs of Lines

(a) Intersecting Lines (प्रतिच्छेदी रेखाएँ)

• Two lines are intersecting if they meet or cross each other at a single common point (called the point of intersection).

(b) Transversal Line (तिर्यक रेखा)

• A line that intersects two or more lines at distinct points is called a transversal.

(c) Angles Made by a Transversal with Two Parallel Lines

When a transversal intersects two parallel lines, several special angle pairs are formed:

1. Corresponding Angles (संगत कोण): Angles in the same relative position at each intersection.
   • Rule: Corresponding angles are equal. ($\angle 1 = \angle 5$, $\angle 2 = \angle 6$, etc.)
2. Alternate Interior Angles (अन्तः एकांतर कोण): Pairs of angles on opposite sides of the transversal and inside the two parallel lines.
   • Rule: Alternate interior angles are equal. ($\angle 3 = \angle 5$, $\angle 4 = \angle 6$)
3. Alternate Exterior Angles (बाह्य एकांतर कोण): Pairs of angles on opposite sides of the transversal and outside the two parallel lines.
   • Rule: Alternate exterior angles are equal. ($\angle 1 = \angle 7$, $\angle 2 = \angle 8$)
4. Interior Angles on the same side of the transversal (तिर्यक रेखा के एक ही ओर के अन्तः कोण): Also called co-interior or consecutive interior angles.
   • Rule: Their sum is supplementary ($180^\circ$). ($\angle 3 + \angle 6 = 180^\circ$ and $\angle 4 + \angle 5 = 180^\circ$)

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5. Checking for Parallel Lines (समानता की जाँच)

Two lines are parallel if:

• Alternate interior angles are equal, OR
• Corresponding angles are equal, OR
• Interior angles on the same side of the transversal are supplementary ($180^\circ$).