krit.club logo

Parallel and Intersecting Lines - Parallel Illusions

Grade 7CBSE

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

🔑Concepts

•

The Parallel Illusion: Parallel lines are two or more lines in a plane that never meet, no matter how far they are extended. Sometimes, our eyes can be deceived by surrounding patterns (like slanted transversals), making straight parallel lines appear curved or tilted. This is known as an optical illusion.

Zollner illusion showing parallel horizontal lines intersected by short slanted lines, making the horizontal lines appear non-parallel.
•

A Transversal is a line that intersects two or more lines at distinct points. When a transversal intersects two parallel lines, specific angle relationships are formed, such as corresponding angles, alternate interior angles, and co-interior angles.

Two parallel lines l and m intersected by a transversal line t.
•

Alternate Interior Angles: These are pairs of angles on opposite sides of the transversal, lying between the two parallel lines. If l∥ml \parallel m, then these angles are always equal.

•

Co-interior Angles (Consecutive Interior Angles): These are angles on the same side of the transversal and inside the parallel lines. Their sum is always 180∘180^\circ (supplementary).

📐Formulae

If l∥m, then Corresponding Angles are equal.\text{If } l \parallel m, \text{ then } \text{Corresponding Angles are equal.}

If l∥m, then Alternate Interior Angles are equal.\text{If } l \parallel m, \text{ then } \text{Alternate Interior Angles are equal.}

∠A+∠B=180∘ (Co-interior angles)\angle A + \angle B = 180^\circ \text{ (Co-interior angles)}

∠x+∠y=180∘ (Linear Pair of angles)\angle x + \angle y = 180^\circ \text{ (Linear Pair of angles)}

Vertically Opposite Angles are always equal: ∠1=∠3\text{Vertically Opposite Angles are always equal: } \angle 1 = \angle 3

💡Examples

Problem 1:

In the given figure, line l∥ml \parallel m and tt is a transversal. If one of the interior angles is 65∘65^\circ, find the measure of its co-interior angle xx.

Solution:

Since l∥ml \parallel m, the sum of co-interior angles is 180∘180^\circ.

65∘+x=180∘65^\circ + x = 180^\circ x=180∘−65∘x = 180^\circ - 65^\circ x=115∘x = 115^\circ

Explanation:

We use the property that interior angles on the same side of a transversal are supplementary when the lines are parallel.

Problem 2:

Two lines pp and qq are cut by a transversal rr. If a pair of alternate interior angles are (3x+10)∘(3x + 10)^\circ and (2x+40)∘(2x + 40)^\circ, for what value of xx will p∥qp \parallel q?

Solution:

For p∥qp \parallel q, the alternate interior angles must be equal:

(3x+10)=(2x+40)(3x + 10) = (2x + 40) 3x−2x=40−103x - 2x = 40 - 10 x=30x = 30

Explanation:

By the property of parallel lines, alternate interior angles are equal. Setting the expressions equal to each other allows us to solve for xx.

Problem 3:

If ∠1\angle 1 and ∠2\angle 2 form a linear pair and ∠1=120∘\angle 1 = 120^\circ, calculate the value of ∠2\angle 2 using vertical arithmetic subtraction.

Solution:

Since they form a linear pair, ∠1+∠2=180∘\angle 1 + \angle 2 = 180^\circ. ∠2=180∘−120∘\angle 2 = 180^\circ - 120^\circ 180−12060\begin{array}{r} 180 \\ - 120 \\ \hline 60 \end{array} So, ∠2=60∘\angle 2 = 60^\circ.

Explanation:

A linear pair consists of adjacent angles formed by intersecting lines whose sum is always 180180 degrees.

Problem 4:

In the following figure, line ABAB is parallel to line CDCD (AB∥CDAB \parallel CD). If ∠1=75∘\angle 1 = 75^\circ, find the value of ∠2\angle 2.

Parallel lines AB and CD with a transversal. Angle 1 is at the intersection with AB and Angle 2 is at the intersection with CD.

Solution:

  1. Identify the relationship: ∠1\angle 1 and ∠2\angle 2 are Alternate Interior Angles because they are on opposite sides of the transversal and inside the parallel lines.
  2. Apply the property: Since AB∥CDAB \parallel CD, the alternate interior angles must be equal.
  3. Therefore, ∠2=∠1=75∘\angle 2 = \angle 1 = 75^\circ.

Explanation:

When a transversal cuts two parallel lines, the Z-shape formed identifies alternate interior angles, which are congruent.

Problem 5:

Given m∥nm \parallel n. If ∠x\angle x and ∠y\angle y are co-interior angles such that ∠x=110∘\angle x = 110^\circ, calculate the value of ∠y\angle y using vertical subtraction.

Parallel lines m and n with a perpendicular transversal. Angles x and y are shown on the same side between the lines.

Solution:

  1. Identify the relationship: Since m∥nm \parallel n, the sum of co-interior angles is 180∘180^\circ.
  2. Equation: ∠x+∠y=180∘\angle x + \angle y = 180^\circ.
  3. Substitute the value: 110∘+∠y=180∘110^\circ + \angle y = 180^\circ.
  4. Calculate ∠y\angle y: ∠y=180∘−110∘\angle y = 180^\circ - 110^\circ. 180−11070\begin{array}{r} 180 \\ - 110 \\ \hline 70 \end{array}
  5. ∠y=70∘\angle y = 70^\circ.

Explanation:

Co-interior angles are supplementary when lines are parallel. We subtract the given angle from 180∘180^\circ to find the unknown angle.