krit.club logo

Parallel and Intersecting Lines - Transversals

Grade 7CBSE

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

🔑Concepts

•

A transversal is a line that intersects two or more lines at distinct points. When a transversal intersects two lines, eight angles are formed.

Transversal p intersecting two lines l and m showing 8 angles.
•

Corresponding Angles are in the same relative position at each intersection where a straight line crosses two others. If the lines are parallel, these angles are equal (e.g., ∠2=∠6\angle 2 = \angle 6).

•

Alternate Interior Angles are pairs of angles on opposite sides of the transversal and between the two lines. When lines are parallel, these angles form a 'Z' shape and are equal (e.g., ∠3=∠5\angle 3 = \angle 5).

•

Interior Angles on the Same Side (Co-interior angles) are supplementary (180∘180^\circ) if the lines being intersected are parallel. They form a 'C' or 'U' shape.

•

To check if two lines are parallel, verify if: 1. Any pair of corresponding angles are equal, 2. Any pair of alternate interior angles are equal, or 3. Interior angles on the same side are supplementary.

📐Formulae

If l∥m, then Corresponding Angles are equal: ∠1=∠5\text{If } l \parallel m, \text{ then Corresponding Angles are equal: } \angle 1 = \angle 5

If l∥m, then Alternate Interior Angles are equal: ∠3=∠6\text{If } l \parallel m, \text{ then Alternate Interior Angles are equal: } \angle 3 = \angle 6

If l∥m, then Interior Angles on the same side (Co-interior): ∠3+∠5=180∘\text{If } l \parallel m, \text{ then Interior Angles on the same side (Co-interior): } \angle 3 + \angle 5 = 180^\circ

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

Vertically Opposite Angles: ∠1=∠4\text{Vertically Opposite Angles: } \angle 1 = \angle 4

💡Examples

Problem 1:

In the given figure, l∥ml \parallel m and tt is a transversal. If one of the angles is 70∘70^\circ, find the measure of its alternate interior angle.

Solution:

70∘70^\circ

Explanation:

Since the lines ll and mm are parallel, the property of transversals states that the pairs of alternate interior angles are equal. Therefore, the alternate interior angle to the given 70∘70^\circ angle is also 70∘70^\circ.

Problem 2:

Two parallel lines are intersected by a transversal. If the measure of one interior angle is (2x+10)∘(2x + 10)^\circ and the other interior angle on the same side of the transversal is (3x+20)∘(3x + 20)^\circ, find the value of xx.

Solution:

x=30x = 30

Explanation:

Interior angles on the same side of a transversal for parallel lines are supplementary. Therefore: (2x+10)+(3x+20)=180(2x + 10) + (3x + 20) = 180 5x+30=1805x + 30 = 180 5x=1505x = 150 x=1505=30x = \frac{150}{5} = 30

Problem 3:

If ∠1=120∘\angle 1 = 120^\circ and ∠5=120∘\angle 5 = 120^\circ are corresponding angles formed by a transversal cutting two lines pp and qq, are the lines pp and qq parallel?

Solution:

Yes, p∥qp \parallel q.

Explanation:

According to the converse of the corresponding angles axiom, if a transversal intersects two lines such that a pair of corresponding angles are equal (120∘=120∘120^\circ = 120^\circ), then the two lines must be parallel.

Problem 4:

In the following figure, line p∥qp \parallel q and line rr is a transversal. If ∠a=110∘\angle a = 110^\circ, find the measure of ∠b\angle b.

Two parallel lines p and q with a transversal r showing co-interior angles a and b.

Solution:

  1. ∠a\angle a and ∠b\angle b are interior angles on the same side of the transversal rr.
  2. Since p∥qp \parallel q, these angles are supplementary.
  3. ∠a+∠b=180∘\angle a + \angle b = 180^\circ
  4. 110∘+∠b=180∘110^\circ + \angle b = 180^\circ
  5. ∠b=180∘−110∘=70∘\angle b = 180^\circ - 110^\circ = 70^\circ

Explanation:

Because the lines are parallel, the sum of the co-interior angles must be 180∘180^\circ.

Problem 5:

Given AB∥CDAB \parallel CD, find the value of xx if the alternate interior angles are 3x−15∘3x - 15^\circ and 2x+10∘2x + 10^\circ.

Parallel lines AB and CD with a transversal showing alternate interior angles labeled with algebraic expressions.

Solution:

  1. Since AB∥CDAB \parallel CD, alternate interior angles are equal.
  2. 3x−15=2x+103x - 15 = 2x + 10
  3. Subtract 2x2x from both sides: x−15=10x - 15 = 10
  4. Add 1515 to both sides: x=25x = 25

Explanation:

The property of parallel lines states that alternate interior angles formed by a transversal are always equal.