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

Grade 7CBSE

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

🔑Concepts

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Intersecting Lines: Two lines ll and mm are called intersecting lines if they have a common point. This common point is called the point of intersection. When two lines intersect, they form four angles, and the vertically opposite angles are equal.

Two intersecting lines l and m forming four angles at point O.
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Transversal: A line that intersects two or more lines at distinct points is called a transversal. In the figure, line pp is a transversal to lines ll and mm. It creates 8 specific angles: Interior, Exterior, Corresponding, Alternate Interior, and Alternate Exterior angles.

A transversal line p crossing two lines l and m, forming eight angles.
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Parallel Lines: If two parallel lines are cut by a transversal, then: (i) each pair of corresponding angles are equal, (ii) each pair of alternate interior angles are equal, and (iii) each pair of interior angles on the same side of the transversal (co-interior) are supplementary (sum to 180∘180^\circ).

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Checking for Parallelism: To prove two lines are parallel, we can check if any one of the following is true: (i) a pair of corresponding angles is equal, (ii) a pair of alternate interior angles is equal, or (iii) co-interior angles add up to 180∘180^\circ.

📐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

If l∥m, then ∠Corresponding1=∠Corresponding2\text{If } l \parallel m, \text{ then } \angle \text{Corresponding}_1 = \angle \text{Corresponding}_2

If l∥m, then ∠Alternate Interior1=∠Alternate Interior2\text{If } l \parallel m, \text{ then } \angle \text{Alternate Interior}_1 = \angle \text{Alternate Interior}_2

If l∥m, then ∠Co-interior1+∠Co-interior2=180∘\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.

Problem 3:

In the given figure, line ll is parallel to mm (l∥ml \parallel m). Find the value of yy if the two interior angles on the same side of the transversal nn are (3y+20)∘(3y + 20)^\circ and (2y−10)∘(2y - 10)^\circ.

Parallel lines l and m with transversal n showing co-interior angles (3y+20) and (2y-10).

Solution:

Given l∥ml \parallel m, the sum of co-interior angles is 180∘180^\circ. (3y+20)+(2y−10)=180(3y + 20) + (2y - 10) = 180 5y+10=1805y + 10 = 180 5y=1705y = 170 y=1705y = \frac{170}{5} y=34y = 34

Explanation:

Since the lines are parallel, the angles interior to the parallel lines and on the same side of the transversal must be supplementary. We set up an equation summing the two expressions to 180∘180^\circ and solve for yy.

Problem 4:

Determine if line pp is parallel to line qq if the alternate interior angles shown in the figure are 115∘115^\circ and 115∘115^\circ.

Two lines p and q with a transversal showing two equal alternate interior angles of 115 degrees.

Solution:

In the figure, the two angles marked are alternate interior angles formed by transversal tt with lines pp and qq. Since the measure of both alternate interior angles is 115∘115^\circ, they are equal. According to the property of parallel lines, if alternate interior angles are equal, the lines must be parallel. Therefore, p∥qp \parallel q.

Explanation:

We use the converse of the alternate interior angle property. Since the given pair of alternate interior angles are equal, it satisfies the condition for lines pp and qq to be parallel.