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Lines and Angles - Angles made by a Transversal

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. Visually, if you have two horizontal lines ll and mm, a transversal pp is the slanted line that cuts across both, creating two separate intersection points and eight resulting angles.

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Interior and Exterior Angles: The angles that lie in the region between the two lines ll and mm are called Interior Angles. The angles that lie outside this region are called Exterior Angles. Think of the space between the two lines as a 'sandwich'; the 'filling' contains the interior angles, while the 'bread' on the outside contains the exterior angles.

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Corresponding Angles: These are pairs of angles that occupy the same relative position at each intersection where a transversal crosses two lines. For example, the top-right angle at the first intersection and the top-right angle at the second intersection are corresponding. If the two lines are parallel, these angles are equal (∠1=∠2\angle 1 = \angle 2).

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Alternate Interior Angles: These are pairs of angles on opposite sides of the transversal but inside the two lines. Visually, they form a 'Z' or 'N' shape. When the transversal cuts two parallel lines, these alternate interior angles are equal. For instance, the angle on the inner-left of the first intersection equals the angle on the inner-right of the second intersection.

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Alternate Exterior Angles: These are pairs of angles on opposite sides of the transversal and outside the two lines. If the lines being intersected are parallel, then these angles are equal. Visually, if you pick the 'outermost' angle on the top-left, its alternate exterior partner is the 'outermost' angle on the bottom-right.

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Interior Angles on the Same Side of the Transversal: Also known as co-interior or consecutive interior angles, these lie between the two lines and on the same side of the transversal. Visually, they form a 'C' or 'U' shape. If the lines are parallel, these two angles are supplementary, meaning their sum is exactly 180∘180^{\circ}.

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Conditions for Parallelism: You can prove two lines are parallel if a transversal creates: (a) equal corresponding angles, (b) equal alternate interior angles, or (c) co-interior angles that add up to 180∘180^{\circ}.

📐Formulae

If l∥ml \parallel m, then Corresponding Angles: ∠1=∠5\angle 1 = \angle 5

If l∥ml \parallel m, then Alternate Interior Angles: ∠3=∠6\angle 3 = \angle 6

If l∥ml \parallel m, then Alternate Exterior Angles: ∠1=∠8\angle 1 = \angle 8

If l∥ml \parallel m, then Co-interior Angles: ∠3+∠5=180∘\angle 3 + \angle 5 = 180^{\circ}

Linear Pair Equation: ∠1+∠2=180∘\angle 1 + \angle 2 = 180^{\circ} (angles on a straight line)

Vertically Opposite Angles: ∠1=∠3\angle 1 = \angle 3 (angles opposite each other at a single vertex)

💡Examples

Problem 1:

In a 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 value of its corresponding co-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. Let the alternate interior angle equal to ∠A\angle A be ∠B\angle B. Since l∥ml \parallel m, ∠B=∠A=75∘\angle B = \angle A = 75^{\circ}.
  3. Now, consider the co-interior angle to ∠B\angle B, let's call it ∠C\angle C.
  4. We know that for parallel lines, co-interior angles are supplementary: ∠B+∠C=180∘\angle B + \angle C = 180^{\circ}.
  5. Substitute the value: 75∘+∠C=180∘75^{\circ} + \angle C = 180^{\circ}.
  6. Solve for ∠C\angle C: ∠C=180∘−75∘=105∘\angle C = 180^{\circ} - 75^{\circ} = 105^{\circ}.

Explanation:

This problem uses two properties of parallel lines: first, that alternate interior angles are equal, and second, that co-interior angles sum to 180∘180^{\circ}.

Problem 2:

Two parallel lines are cut by a transversal. If a pair of corresponding angles are represented by the expressions (2x+15)∘(2x + 15)^{\circ} and (3x−10)∘(3x - 10)^{\circ}, find the value of xx.

Solution:

  1. Since the lines are parallel, corresponding angles must be equal in measure.
  2. Set up the equation: (2x+15)=(3x−10)(2x + 15) = (3x - 10).
  3. Subtract 2x2x from both sides: 15=x−1015 = x - 10.
  4. Add 1010 to both sides: 15+10=x15 + 10 = x.
  5. Therefore, x=25x = 25.

Explanation:

The solution relies on the fundamental property that corresponding angles are equal when a transversal intersects parallel lines. We translate this geometric property into an algebraic equation to solve for the unknown variable.