Lines And Angles

Geometry की foundation इन्हीं terms पर based है। इन्हें अच्छे से समझना बहुत ज़रूरी है।

• Point: A dimensionless location in space. Represented by a dot. No length, breadth, or height.
• Line: A straight path that extends infinitely in both directions. No endpoints. Denoted by \( \overleftrightarrow{AB} \) or a small letter like \(l\).
• Line Segment: A part of a line with two distinct endpoints. Has a definite length. Denoted by \( \overline{AB} \).
• Ray: A part of a line with one endpoint and extends infinitely in one direction. Denoted by \( \overrightarrow{AB} \).
• Collinear Points: Three or more points that lie on the same line.
• Non-collinear Points: Points that do not lie on the same line.
• Intersecting Lines: Two lines that share exactly one common point. This common point is called the point of intersection.
• Parallel Lines: Two lines in a plane that never intersect, no matter how far they are extended. The perpendicular distance between them is always constant. Denoted by \(l \parallel m\).
• Perpendicular Lines: Two lines that intersect to form a right angle (90°). Denoted by \(l \perp m\).
• Angle: Formed when two rays originate from the same endpoint. The rays are called the arms of the angle, and the common endpoint is called the vertex. Denoted by \( \angle ABC \) or \( \angle B \) or \( \angle 1 \).

Angles को उनके measure के हिसाब से classify किया जाता है।

• Acute Angle: An angle whose measure is between 0° and 90°. (e.g., 30°, 60°)
• Right Angle: An angle whose measure is exactly 90°. Represented by a square symbol at the vertex.
• Obtuse Angle: An angle whose measure is between 90° and 180°. (e.g., 120°, 150°)
• Straight Angle: An angle whose measure is exactly 180°. Forms a straight line.
• Reflex Angle: An angle whose measure is between 180° and 360°. (e.g., 210°, 300°)
• Complete Angle: An angle whose measure is exactly 360°. A full rotation.

Angles के pairs और उनके properties exam में बहुत पूछे जाते हैं।

• Complementary Angles: Two angles are complementary if their sum is 90°. Each angle is the complement of the other. (e.g., 30° and 60°)
• Supplementary Angles: Two angles are supplementary if their sum is 180°. Each angle is the supplement of the other. (e.g., 70° and 110°)
• Adjacent Angles: Two angles are adjacent if they have:
• A common vertex.
• A common arm.
• Their non-common arms are on opposite sides of the common arm.

(e.g., \( \angle AOB \) and \( \angle BOC \) with common vertex O and common arm OB).

• Linear Pair of Angles: A pair of adjacent angles whose non-common arms are opposite rays (form a straight line). Their sum is always 180°. A linear pair is always supplementary.
• Vertically Opposite Angles: When two lines intersect, they form two pairs of vertically opposite angles. These angles are always equal.
• If lines AB and CD intersect at O, then \( \angle AOC = \angle BOD \) and \( \angle AOD = \angle BOC \).
• Proof of Vertically Opposite Angles being equal:

1. \( \angle AOC + \angle AOD = 180° \) (Linear Pair)
2. \( \angle AOD + \angle BOD = 180° \) (Linear Pair)
3. From (1) and (2), \( \angle AOC + \angle AOD = \angle AOD + \angle BOD \)
4. Subtracting \( \angle AOD \) from both sides, we get \( \angle AOC = \angle BOD \).
5. Similarly, \( \angle AOD = \angle BOC \) can be proved.

जब एक transversal line दो parallel lines को cut करती है, तो कई special angle relationships बनते हैं।

• Transversal: A line that intersects two or more lines at distinct points.
• When a transversal intersects two parallel lines, it forms 8 angles. These angles have specific relationships:
• Corresponding Angles: Angles in the same relative position at each intersection. They are equal.
• Pairs: \( \angle 1 = \angle 5 \), \( \angle 2 = \angle 6 \), \( \angle 3 = \angle 7 \), \( \angle 4 = \angle 8 \).
• Alternate Interior Angles: Angles on opposite sides of the transversal and between the parallel lines. They are equal.
• Pairs: \( \angle 3 = \angle 6 \), \( \angle 4 = \angle 5 \).
• Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the parallel lines. They are equal.
• Pairs: \( \angle 1 = \angle 8 \), \( \angle 2 = \angle 7 \).
• Consecutive Interior Angles (or Co-interior/Allied Angles): Angles on the same side of the transversal and between the parallel lines. They are supplementary (sum is 180°).
• Pairs: \( \angle 4 + \angle 6 = 180° \), \( \angle 3 + \angle 5 = 180° \).

• Axioms and Theorems related to Parallel Lines:
• Axiom 6.3 (Corresponding Angles Axiom): If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. (This is an axiom, so it's assumed true).
• Theorem 6.2: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
• Theorem 6.3: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. (Converse of Theorem 6.2).
• Theorem 6.4: If a transversal intersects two parallel lines, then each pair of consecutive interior angles is supplementary.
• Theorem 6.5: If a transversal intersects two lines such that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
• Theorem 6.6: Lines which are parallel to the same line are parallel to each other. (If \(l \parallel m\) and \(m \parallel n\), then \(l \parallel n\)).

Triangle के angles का sum हमेशा fixed होता है। ये property बहुत useful है।

• Theorem 6.7 (Angle Sum Property of a Triangle): The sum of the angles of a triangle is 180°.
• For any triangle \( \triangle ABC \), \( \angle A + \angle B + \angle C = 180° \).

• Proof of Angle Sum Property:

1. Consider a triangle \( \triangle ABC \).
2. Draw a line PQ parallel to BC passing through vertex A.
3. Since PQ is a straight line, \( \angle PAB + \angle BAC + \angle CAQ = 180° \) (Angles on a straight line).
4. Since PQ \( \parallel \) BC and AB is a transversal, \( \angle PAB = \angle ABC \) (Alternate interior angles).
5. Since PQ \( \parallel \) BC and AC is a transversal, \( \angle CAQ = \angle ACB \) (Alternate interior angles).
6. Substitute (4) and (5) into (3):

\( \angle ABC + \angle BAC + \angle ACB = 180° \).

1. Therefore, \( \angle A + \angle B + \angle C = 180° \).

Triangle के exterior angles की भी एक special property होती है।

• Theorem 6.8 (Exterior Angle Property): If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
• Consider \( \triangle ABC \). If side BC is produced to D, then \( \angle ACD \) is the exterior angle.
• The interior opposite angles are \( \angle BAC \) and \( \angle ABC \).
• According to the property, \( \angle ACD = \angle BAC + \angle ABC \).

• Proof of Exterior Angle Property:

1. In \( \triangle ABC \), we know that \( \angle BAC + \angle ABC + \angle ACB = 180° \) (Angle Sum Property).
2. Also, \( \angle ACB + \angle ACD = 180° \) (Linear Pair).
3. From (1) and (2), we can equate the sums:

\( \angle BAC + \angle ABC + \angle ACB = \angle ACB + \angle ACD \).

1. Subtracting \( \angle ACB \) from both sides, we get:

\( \angle BAC + \angle ABC = \angle ACD \).

1. Thus, the exterior angle is equal to the sum of the two interior opposite angles.