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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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A transversal is a line that intersects two or more lines at distinct points. When a transversal intersects two parallel lines, specific pairs of angles are formed: Corresponding, Alternate, and Interior angles.

Transversal line t intersecting two parallel lines l and m.
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Corresponding Angles: These are pairs of angles in similar positions at each intersection. If the lines are parallel, corresponding angles are equal, such as ∠1=∠3\angle 1 = \angle 3.

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Alternate Interior Angles: These are pairs of angles on opposite sides of the transversal and between the two lines. For parallel lines, they are equal (e.g., ∠2=∠3\angle 2 = \angle 3).

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Co-interior (Consecutive Interior) Angles: These are pairs of angles on the same side of the transversal and between the two lines. For parallel lines, they are supplementary, meaning their sum is 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.

Problem 3:

In the given figure, p∥qp \parallel q. If ∠1=120∘\angle 1 = 120^\circ, find the measure of ∠5\angle 5 and ∠8\angle 8.

Parallel lines p and q with transversal showing angles 1, 5, and 8.

Solution:

∠1=120∘\angle 1 = 120^\circ Since p∥qp \parallel q, ∠5\angle 5 and ∠1\angle 1 are corresponding angles. ∠5=∠1=120∘\angle 5 = \angle 1 = 120^\circ ∠5\angle 5 and ∠8\angle 8 form a linear pair. ∠5+∠8=180∘\angle 5 + \angle 8 = 180^\circ 120∘+∠8=180∘120^\circ + \angle 8 = 180^\circ ∠8=180∘−120∘=60∘\angle 8 = 180^\circ - 120^\circ = 60^\circ

Explanation:

We use the property that corresponding angles of parallel lines are equal to find ∠5\angle 5, and the linear pair property to find ∠8\angle 8.

Problem 4:

Given AB∥CDAB \parallel CD, find the value of yy if the interior angles on the same side of the transversal are (4y+10)∘(4y + 10)^\circ and (2y−10)∘(2y - 10)^\circ.

Parallel lines AB and CD with co-interior angles labeled.

Solution:

Interior angles on the same side of a transversal of parallel lines are supplementary. (4y+10)+(2y−10)=180(4y + 10) + (2y - 10) = 180 4y+2y+10−10=1804y + 2y + 10 - 10 = 180 6y=1806y = 180 y=1806y = \frac{180}{6} y=30y = 30

Explanation:

Co-interior angles between parallel lines always sum to 180∘180^\circ. We set up an equation with the given algebraic expressions and solve for yy.