krit.club logo

Lines and Angles - Intersecting Lines and Non-intersecting Lines

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Intersecting Lines: Two lines are called intersecting if they share exactly one common point. Visually, these lines form an 'X' shape, and the common point where they meet is known as the point of intersection.

•

Non-intersecting or Parallel Lines: Lines in a plane that do not meet at any point, no matter how far they are extended, are called parallel lines. Visually, they look like the two rails of a straight railway track, and the perpendicular distance between them remains constant everywhere.

•

Vertically Opposite Angles: When two lines intersect at a point, they form two pairs of angles that are opposite to each other at the vertex. These are called vertically opposite angles and are always equal in measure. Visually, in an 'X' shape, the top angle equals the bottom angle, and the left angle equals the right angle.

•

Linear Pair of Angles: A linear pair is a pair of adjacent angles whose non-common arms form a straight line. According to the Linear Pair Axiom, the sum of these two angles is always 180∘180^\circ. Visually, if a ray stands on a straight line, the two angles on either side of the ray form a straight horizontal base.

•

Perpendicular Lines: These are a special case of intersecting lines where the angle between the two lines is exactly 90∘90^\circ. Visually, they form a perfect 'L' or '+' shape, indicating a right-angle intersection.

•

Adjacent Angles: Two angles are adjacent if they have a common vertex, a common arm, and their non-common arms are on different sides of the common arm. Visually, they are 'side-by-side' angles sharing a boundary.

•

Perpendicular Distance: This concept applies specifically to parallel lines; it is the shortest distance between two lines. For non-intersecting lines, this distance is the same at every point along the lines.

•

Axiom of Straight Lines: If the sum of two adjacent angles is 180∘180^\circ, then the non-common arms of the angles form a straight line. Visually, this confirms that a flat, unbroken surface represents a 180∘180^\circ angle.

📐Formulae

Linear Pair: ∠1+∠2=180∘\angle 1 + \angle 2 = 180^\circ

Vertically Opposite Angles: ∠AOC=∠BOD\angle AOC = \angle BOD and ∠AOD=∠BOC\angle AOD = \angle BOC (for lines ABAB and CDCD intersecting at OO)

Sum of angles around a point: ∑θ=360∘\sum \theta = 360^\circ

Perpendicular Condition: l⊥m  ⟹  ∠(l,m)=90∘l \perp m \implies \angle(l, m) = 90^\circ

Parallel Condition: l∥m  ⟹  Distance (d) is constantl \parallel m \implies \text{Distance } (d) \text{ is constant}

💡Examples

Problem 1:

Two lines ABAB and CDCD intersect at point OO. If ∠AOC=55∘\angle AOC = 55^\circ, find the measures of ∠BOC\angle BOC, ∠BOD\angle BOD, and ∠AOD\angle AOD.

Solution:

  1. Since ABAB is a straight line and ray OCOC stands on it, ∠AOC\angle AOC and ∠BOC\angle BOC form a linear pair. Therefore, ∠AOC+∠BOC=180∘\angle AOC + \angle BOC = 180^\circ.
  2. Substitute the given value: 55∘+∠BOC=180∘  ⟹  ∠BOC=180∘−55∘=125∘55^\circ + \angle BOC = 180^\circ \implies \angle BOC = 180^\circ - 55^\circ = 125^\circ.
  3. ∠BOD\angle BOD and ∠AOC\angle AOC are vertically opposite angles, so ∠BOD=∠AOC=55∘\angle BOD = \angle AOC = 55^\circ.
  4. ∠AOD\angle AOD and ∠BOC\angle BOC are vertically opposite angles, so ∠AOD=∠BOC=125∘\angle AOD = \angle BOC = 125^\circ.

Explanation:

This problem uses the Linear Pair Axiom to find the adjacent supplement and the property of Vertically Opposite Angles to find the angles across the intersection.

Problem 2:

In a figure, lines XYXY and MNMN intersect at OO. If ∠POY=90∘\angle POY = 90^\circ and a:b=2:3a : b = 2 : 3 (where a=∠POMa = \angle POM and b=∠MOXb = \angle MOX), find the value of c=∠XONc = \angle XON.

Solution:

  1. Since XYXY is a line, ∠XOP+∠POY=180∘\angle XOP + \angle POY = 180^\circ. Given ∠POY=90∘\angle POY = 90^\circ, then ∠XOP=180∘−90∘=90∘\angle XOP = 180^\circ - 90^\circ = 90^\circ.
  2. We know ∠XOP=a+b\angle XOP = a + b, so a+b=90∘a + b = 90^\circ.
  3. Given a:b=2:3a : b = 2 : 3, let a=2xa = 2x and b=3xb = 3x. Then 2x+3x=90∘  ⟹  5x=90∘  ⟹  x=18∘2x + 3x = 90^\circ \implies 5x = 90^\circ \implies x = 18^\circ.
  4. Calculate bb: b=3×18∘=54∘b = 3 \times 18^\circ = 54^\circ.
  5. Since MNMN is a straight line, ∠MOX\angle MOX and ∠XON\angle XON form a linear pair. Thus, b+c=180∘b + c = 180^\circ.
  6. Substitute bb: 54∘+c=180∘  ⟹  c=180∘−54∘=126∘54^\circ + c = 180^\circ \implies c = 180^\circ - 54^\circ = 126^\circ.

Explanation:

The solution first uses the linear pair property on line XYXY to isolate the sum of aa and bb, then uses ratios to find their specific values, and finally applies the linear pair property on line MNMN to find cc.