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Lines and Angles - Parallel Lines and a Transversal

Grade 9CBSE

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

🔑Concepts

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Transversal Line: A line that intersects two or more lines at distinct points is known as a transversal. Visually, if you have two horizontal parallel lines, a transversal appears as a slanted line passing through both, creating two distinct intersection points and a total of eight angles.

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Corresponding Angles: When two parallel lines are cut by a transversal, the angles in matching corners are called corresponding angles. These are equal in measure and can be identified by looking for an 'F-shape' (oriented in any direction) in the diagram, where the angles lie in the same relative position at each intersection.

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Alternate Interior Angles: These are pairs of angles that lie between the two parallel lines but on opposite sides of the transversal. When the lines are parallel, these angles are equal. You can visualize this as a 'Z-shape' or 'N-shape', where the angles are tucked into the inner corners of the 'Z'.

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Alternate Exterior Angles: These angles are located outside the parallel lines and on opposite sides of the transversal. If the lines being intersected are parallel, then the alternate exterior angles are equal to each other.

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Interior Angles on the Same Side (Co-interior Angles): Also referred to as consecutive interior angles, these lie between the parallel lines and on the same side of the transversal. For parallel lines, these angles are supplementary, meaning their sum is 180∘180^\circ. Visually, they form a 'C' or 'U' shape between the lines.

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Converse Properties: The relationship is bidirectional; if a transversal intersects two lines such that any pair of corresponding angles are equal, or alternate interior angles are equal, or co-interior angles are supplementary, then the two lines must be parallel.

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Linear Pair and Vertically Opposite Angles: While not unique to parallel lines, these are essential for solving problems. Angles on a straight line add up to 180∘180^\circ (Linear Pair), and angles opposite each other at an intersection are equal (Vertically Opposite).

📐Formulae

If l∥ml \parallel m, then ∠extCorresponding1=∠extCorresponding2\angle ext{Corresponding}_1 = \angle ext{Corresponding}_2

If l∥ml \parallel m, then ∠extAlt.Interior1=∠extAlt.Interior2\angle ext{Alt. Interior}_1 = \angle ext{Alt. Interior}_2

Sum of Co-interior Angles: ∠extInt1+∠extInt2=180∘\angle ext{Int}_1 + \angle ext{Int}_2 = 180^\circ (when lines are parallel)

Sum of Angles on a Straight Line: ∠a+∠b=180∘\angle a + \angle b = 180^\circ

Vertically Opposite Angles: ∠1=∠3\angle 1 = \angle 3 and ∠2=∠4\angle 2 = \angle 4 at any intersection

💡Examples

Problem 1:

In the figure, line AB∥CDAB \parallel CD and a transversal PQPQ intersects them. If one of the interior angles on the same side of the transversal is (3x+20)∘(3x + 20)^\circ and the other is (2x−10)∘(2x - 10)^\circ, find the value of xx.

Solution:

  1. Since AB∥CDAB \parallel CD, the sum of interior angles on the same side of the transversal (co-interior angles) is 180∘180^\circ.\n2. Write the equation: (3x+20)+(2x−10)=180(3x + 20) + (2x - 10) = 180.\n3. Combine like terms: 5x+10=1805x + 10 = 180.\n4. Subtract 1010 from both sides: 5x=1705x = 170.\n5. Divide by 55: x=1705=34x = \frac{170}{5} = 34.

Explanation:

The solution uses the Co-interior Angle Theorem which states that consecutive interior angles are supplementary when lines are parallel.

Problem 2:

Two parallel lines are intersected by a transversal. If a pair of alternate interior angles are given by 55∘55^\circ and (2y−5)∘(2y - 5)^\circ, find the value of yy.

Solution:

  1. Identify the relationship: Alternate interior angles are equal when lines are parallel.\n2. Set up the equation: 2y−5=552y - 5 = 55.\n3. Add 55 to both sides: 2y=602y = 60.\n4. Divide by 22: y=30y = 30.

Explanation:

This problem relies on the property that alternate interior angles (the 'Z-shape' angles) have the same measure when lines are parallel.