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Geometry - Parallel Lines and Transversal: Corresponding Angles, Alternate Angles, Interior Angles

Grade 7ICSE

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

🔑Concepts

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Parallel Lines and Transversal: Parallel lines are lines in the same plane that never intersect, denoted by l∥ml \parallel m. A transversal is a line that intersects two or more parallel lines at distinct points. Visually, a transversal acts as a bridge crossing over the parallel tracks, creating eight distinct angles at the two points of intersection.

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Corresponding Angles: These angles are in the same relative position at each intersection where a straight line crosses two others. Visually, they form an 'F' shape (either forwards, backwards, upside down, or both). If the two lines are parallel, the corresponding angles are equal, such as ∠1=∠5\angle 1 = \angle 5.

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Alternate Interior Angles: These are pairs of angles on opposite sides of the transversal and between the two parallel lines. Visually, they form a 'Z' or 'N' shape. When the lines are parallel, these interior angles are equal in measure.

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Alternate Exterior Angles: These are pairs of angles on opposite sides of the transversal that lie outside the parallel lines. If the lines intersected by the transversal are parallel, then these alternate exterior angles are equal.

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Co-interior Angles (Consecutive Interior Angles): These are pairs of angles located on the same side of the transversal and between the two parallel lines. Visually, they form a 'C' or 'U' shape. When lines are parallel, co-interior angles are supplementary, meaning their sum is 180∘180^\circ.

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Vertically Opposite Angles: When the transversal intersects any line, the angles opposite each other at the vertex are called vertically opposite angles. These angles are always equal (e.g., ∠1=∠3\angle 1 = \angle 3) regardless of whether the lines are parallel.

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Linear Pair: A linear pair consists of two adjacent angles formed by the intersection of the transversal and a line, which together form a straight line. The sum of the angles in a linear pair is always 180∘180^\circ.

📐Formulae

If l∥ml \parallel m, then Corresponding Angles: ∠a=∠b\angle a = \angle b

If l∥ml \parallel m, then Alternate Interior/Exterior Angles: ∠x=∠y\angle x = \angle y

If l∥ml \parallel m, then Co-interior Angles: ∠x+∠y=180∘\angle x + \angle y = 180^\circ

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

Vertically Opposite Angles: ∠1=∠2\angle 1 = \angle 2

💡Examples

Problem 1:

In a figure where L1∥L2L_1 \parallel L_2 and a transversal TT intersects them, one of the alternate interior angles is given as (3x−15)∘(3x - 15)^\circ and the other is 105∘105^\circ. Find the value of xx.

Solution:

  1. Since the lines are parallel, alternate interior angles must be equal.
  2. Set up the equation: 3x−15=1053x - 15 = 105
  3. Add 1515 to both sides: 3x=105+153x = 105 + 15
  4. 3x=1203x = 120
  5. Divide by 33: x=1203x = \frac{120}{3}
  6. x=40x = 40

Explanation:

This problem applies the property that alternate interior angles are equal when a transversal intersects parallel lines. We create an algebraic equation to solve for the unknown variable.

Problem 2:

Two parallel lines are cut by a transversal. If the interior angles on the same side of the transversal (co-interior angles) are in the ratio 2:32:3, find the measure of the larger angle.

Solution:

  1. Let the two co-interior angles be 2x2x and 3x3x.
  2. We know that co-interior angles are supplementary: 2x+3x=180∘2x + 3x = 180^\circ.
  3. Combine like terms: 5x=180∘5x = 180^\circ.
  4. Solve for xx: x=180∘5=36∘x = \frac{180^\circ}{5} = 36^\circ.
  5. Find the larger angle: 3x=3×36∘=108∘3x = 3 \times 36^\circ = 108^\circ.

Explanation:

We use the property that co-interior angles sum to 180∘180^\circ to establish a linear equation based on the given ratio, then calculate the specific angle requested.