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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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A transversal is a line that intersects two or more lines at distinct points. When a transversal line nn cuts across two lines ll and mm, it creates eight angles, and the relationships between these angles help determine if lines ll and mm are parallel.

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The Corresponding Angles property states that if a transversal intersects two lines such that a pair of corresponding angles are equal, then the lines are parallel. Corresponding angles are in the same relative position at each intersection, forming an 'F' shape (upright, inverted, or mirrored).

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The Alternate Interior Angles property indicates that if a transversal intersects two lines such that a pair of alternate interior angles are equal, the lines are parallel. These angles are located between the two lines on opposite sides of the transversal, forming a 'Z' or 'N' shape.

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The Alternate Exterior Angles property states that if a transversal intersects two lines such that a pair of alternate exterior angles are equal, the lines are parallel. These angles are located outside the two lines on opposite sides of the transversal.

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The Consecutive Interior Angles property (also known as Co-interior angles) states that if a pair of interior angles on the same side of the transversal are supplementary (sum up to 180∘180^\circ), then the lines are parallel. This relationship is often visualized as a 'C' or 'U' shape between the lines.

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The property of Lines Parallel to the Same Line states that if two lines are each parallel to a third line, they are parallel to each other. For example, if line a∥ca \parallel c and line b∥cb \parallel c, then a∥ba \parallel b.

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To check for parallel lines in any geometric figure, look for the 'Z' pattern (Alternate Interior), 'F' pattern (Corresponding), or 'C' pattern (Co-interior). If any one of these specific angle conditions is met, the lines are mathematically proven to be parallel.

📐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.