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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. When a transversal intersects two lines, eight angles are formed.

A transversal line t intersecting two parallel lines l and m, creating eight numbered angles.
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Interior and Exterior Angles: Angles lying between the two lines are called interior angles (∠3,∠4,∠5,∠6\angle 3, \angle 4, \angle 5, \angle 6), while those lying outside are called exterior angles (∠1,∠2,∠7,∠8\angle 1, \angle 2, \angle 7, \angle 8).

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Corresponding Angles: These angles are in the same relative position at each intersection. If the lines are parallel, these angles are equal (e.g., ∠1=∠5\angle 1 = \angle 5, ∠2=∠6\angle 2 = \angle 6).

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Alternate Interior Angles: These angles are on opposite sides of the transversal and between the two lines. If lines are parallel, these angles are equal (e.g., ∠3=∠6\angle 3 = \angle 6, ∠4=∠5\angle 4 = \angle 5).

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Co-interior Angles: Also known as consecutive interior angles, they lie on the same side of the transversal. If the lines are parallel, these angles are supplementary (Sum=180∘Sum = 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.

Problem 3:

In the given figure, p∥qp \parallel q and tt is a transversal. If ∠1=120∘\angle 1 = 120^{\circ}, find the measure of ∠2\angle 2.

Parallel lines p and q intersected by transversal t showing angles 1 and 2.

Solution:

∠1+∠3=180∘ (Linear pair)\angle 1 + \angle 3 = 180^{\circ} \text{ (Linear pair)} ∠3=180∘−120∘=60∘\angle 3 = 180^{\circ} - 120^{\circ} = 60^{\circ} ∠2=∠3 (Alternate interior angles)\angle 2 = \angle 3 \text{ (Alternate interior angles)} ∠2=60∘\angle 2 = 60^{\circ}

Explanation:

Since p∥qp \parallel q, we use the property of alternate interior angles. First, we find the angle adjacent to ∠1\angle 1 using the linear pair property, then equate it to ∠2\angle 2. Alternatively, ∠1\angle 1 and ∠2\angle 2 are co-exterior angles on the same side, which are also supplementary.

Problem 4:

Given AB∥CDAB \parallel CD and a transversal EFEF intersects them. If a pair of co-interior angles are (x+20)∘(x + 20)^{\circ} and (2x−5)∘(2x - 5)^{\circ}, find the value of xx.

Parallel lines AB and CD with co-interior angles labeled as expressions of x.

Solution:

(x+20)∘+(2x−5)∘=180∘ (Co-interior angles are supplementary)(x + 20)^{\circ} + (2x - 5)^{\circ} = 180^{\circ} \text{ (Co-interior angles are supplementary)} 3x+15=1803x + 15 = 180 3x=1653x = 165 x=55x = 55

Explanation:

When two parallel lines are intersected by a transversal, the sum of the interior angles on the same side of the transversal (co-interior angles) is always 180∘180^{\circ}.