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Lines and Angles - Checking for Parallel Lines

Grade 7CBSE

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

🔑Concepts

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Corresponding Angles Criterion: Two lines are parallel if a transversal intersects them such that a pair of corresponding angles are equal. In the diagram, if a=ba = b, then line ll is parallel to line mm.

Diagram showing line n intersecting lines l and m with corresponding angles a and b marked.
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Alternate Interior Angles Criterion: If a transversal intersects two lines such that a pair of alternate interior angles are equal, the lines must be parallel.

Diagram showing alternate interior angles x and y between two lines.
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Interior Angles on the Same Side (Co-interior): If a transversal intersects two lines such that the sum of the interior angles on the same side is 180∘180^\circ (supplementary), then the lines are parallel.

Diagram showing co-interior angles p and q.
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Lines parallel to the same line: If two lines are both parallel to a third line, then they are parallel to each other. If l∥nl \parallel n and m∥nm \parallel n, then l∥ml \parallel m.

📐Formulae

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

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

If ∠Co-interior1+∠Co-interior2=180∘  ⟹  l∥m\text{If } \angle \text{Co-interior}_1 + \angle \text{Co-interior}_2 = 180^\circ \implies l \parallel m

If l∥m and m∥n  ⟹  l∥n\text{If } l \parallel m \text{ and } m \parallel n \implies l \parallel n

💡Examples

Problem 1:

In a figure, a transversal nn cuts two lines ll and mm. If a pair of consecutive interior angles are given as (3x+20)∘(3x + 20)^\circ and (2x−10)∘(2x - 10)^\circ, find the value of xx that would make line ll parallel to line mm, given that their sum must be 160∘160^\circ is incorrect and they should be supplementary.

Solution:

Step 1: For lines ll and mm to be parallel, the sum of the consecutive interior angles must be 180∘180^\circ. Step 2: Set up the equation: (3x+20)+(2x−10)=180(3x + 20) + (2x - 10) = 180. Step 3: Combine like terms: 5x+10=1805x + 10 = 180. Step 4: Subtract 1010 from both sides: 5x=1705x = 170. Step 5: Divide by 55: x=1705=34x = \frac{170}{5} = 34.

Explanation:

We use the property that consecutive interior angles must be supplementary (180∘180^\circ) for the lines to be parallel. Solving the linear equation gives the required value of xx.

Problem 2:

Line nn intersects lines pp and qq. If the alternate interior angles are 75∘75^\circ and (2y−5)∘(2y - 5)^\circ, what value of yy ensures p∥qp \parallel q?

Solution:

Step 1: For lines pp and qq to be parallel, the alternate interior angles must be equal. Step 2: Set up the equation: 2y−5=752y - 5 = 75. Step 3: Add 55 to both sides: 2y=802y = 80. Step 4: Divide by 22: y=40y = 40.

Explanation:

According to the Converse of Alternate Interior Angles Theorem, if the alternate interior angles are equal, the lines are parallel. We equate the two given expressions and solve for the variable.

Problem 3:

In the given figure, transversal tt intersects lines rr and ss. If the interior angles on the same side of the transversal are (2x+15)∘(2x + 15)^\circ and 105∘105^\circ, find the value of xx for which line rr is parallel to line ss.

Transversal t crossing lines r and s with labeled co-interior angles.

Solution:

For r∥sr \parallel s, the sum of interior angles on the same side of the transversal must be 180∘180^\circ. (2x+15)∘+105∘=180∘(2x + 15)^\circ + 105^\circ = 180^\circ 2x+120=1802x + 120 = 180 2x=180−1202x = 180 - 120 2x=602x = 60 x=30x = 30

Explanation:

We use the property that co-interior angles are supplementary for parallel lines. Setting their sum to 180 and solving the linear equation gives the required value of xx.

Problem 4:

Check if line AB∥CDAB \parallel CD if the alternate interior angles formed by transversal EFEF are (4y−10)∘(4y - 10)^\circ and (3y+20)∘(3y + 20)^\circ where y=30y = 30.

Diagram of lines AB and CD with transversal EF showing alternate interior angles in terms of y.

Solution:

Substitute y=30y = 30 into the angle expressions: Angle 1: 4(30)−10=120−10=110∘4(30) - 10 = 120 - 10 = 110^\circ Angle 2: 3(30)+20=90+20=110∘3(30) + 20 = 90 + 20 = 110^\circ Since the alternate interior angles are equal (110∘=110∘110^\circ = 110^\circ), AB∥CDAB \parallel CD.

Explanation:

To check for parallelism, we calculate the numerical values of the alternate interior angles. If they are equal, the lines are parallel by the alternate interior angles converse theorem.