krit.club logo

Lines and Angles - Pairs of Lines (Intersecting, Transversal, Parallel)

Grade 7CBSE

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

🔑Concepts

•

Intersecting Lines and Vertically Opposite Angles: When two lines ll and mm cross each other at a single point, they are called intersecting lines. This intersection creates four angles; the pairs of angles directly opposite each other at the vertex are called vertically opposite angles (forming an 'X' shape) and are always equal in measure.

•

Transversal Line: A line that intersects two or more lines at distinct points is known as a transversal. This configuration typically creates eight angles. Visually, if you have two horizontal lines and a third line cutting through them diagonally, that diagonal line is the transversal.

•

Corresponding Angles: When a transversal intersects two lines, angles that are in the same relative position at each intersection are called corresponding angles. Imagine sliding one intersection point along the transversal onto the other; the angles that overlap are corresponding. If the lines are parallel, these angles are equal.

•

Alternate Interior Angles: These are pairs of angles that lie between the two lines (interior) but on opposite sides of the transversal. They visually form a 'Z' or 'N' shape. When the two lines being intersected are parallel, these alternate interior angles are equal in measure.

•

Alternate Exterior Angles: These are pairs of angles that lie outside the two lines (exterior) and on opposite sides of the transversal. Like alternate interior angles, these are equal if the lines intersected by the transversal are parallel.

•

Interior Angles on the Same Side (Co-interior): These are the interior angles that lie on the same side of the transversal, forming a 'C' or 'U' shape between the lines. If the lines are parallel, these angles are supplementary, meaning their sum is exactly 180∘180^\circ.

•

Parallel Lines: Two lines in a plane that never meet, no matter how far they are extended, are called parallel lines. The perpendicular distance between them remains constant everywhere. When a transversal cuts two parallel lines, specific relationships between corresponding, alternate, and co-interior angles are established.

📐Formulae

∠Linear Pair 1+∠Linear Pair 2=180∘\angle \text{Linear Pair 1} + \angle \text{Linear Pair 2} = 180^\circ

∠Vertically Opposite1=∠Vertically Opposite2\angle \text{Vertically Opposite}_1 = \angle \text{Vertically Opposite}_2

\text{If } l \parallel m, \text{ then } \angle \text{Corresponding}_1 = \angle \text{Corresponding}_2$

\text{If } l \parallel m, \text{ then } \angle \text{Alternate Interior}_1 = \angle \text{Alternate Interior}_2$

\text{If } l \parallel m, \text{ then } \angle \text{Co-interior}_1 + \angle \text{Co-interior}_2 = 180^\circ$

💡Examples

Problem 1:

In the given figure, line ll is parallel to line mm (l∥ml \parallel m) and they are intersected by a transversal tt. If one of the alternate interior angles is 75∘75^\circ, find the measure of its adjacent interior angle on the same side of the transversal.

Solution:

  1. Let the given alternate interior angle be ∠A=75∘\angle A = 75^\circ.
  2. We know that for parallel lines, alternate interior angles are equal, but the question asks for the co-interior angle.
  3. Let the co-interior angle to ∠A\angle A be ∠B\angle B.
  4. According to the property of parallel lines, the sum of interior angles on the same side of the transversal is 180∘180^\circ.
  5. Therefore, ∠A+∠B=180∘\angle A + \angle B = 180^\circ.
  6. 75∘+∠B=180∘75^\circ + \angle B = 180^\circ.
  7. ∠B=180∘−75∘=105∘\angle B = 180^\circ - 75^\circ = 105^\circ.

Explanation:

This problem uses the property that co-interior angles are supplementary when lines are parallel.

Problem 2:

Two lines intersect at a point OO. If one of the angles formed is (2x+10)∘(2x + 10)^\circ and the angle vertically opposite to it is 70∘70^\circ, find the value of xx.

Solution:

  1. Vertically opposite angles are equal when two lines intersect.
  2. We can set up the equation: (2x+10)∘=70∘(2x + 10)^\circ = 70^\circ.
  3. Subtract 1010 from both sides: 2x=70−102x = 70 - 10.
  4. 2x=602x = 60.
  5. Divide by 22: x=602x = \frac{60}{2}.
  6. x=30x = 30.

Explanation:

The solution relies on the fundamental property that vertically opposite angles are always equal.