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Parallel and Intersecting Lines - Perpendicular Lines

Grade 7CBSE

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

🔑Concepts

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Two lines are called perpendicular if they intersect each other at a right angle, which is exactly 90∘90^{\circ}. This is denoted by the symbol ⊥\perp. For example, if line ll is perpendicular to line mm, we write it as l⊥ml \perp m.

Diagram showing two lines l and m intersecting at a 90 degree angle.
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Parallel lines are lines in a plane that never meet, no matter how far they are extended. The perpendicular distance between two parallel lines remains constant everywhere. This is often seen in railway tracks or the opposite edges of a ruler.

Diagram of two parallel lines p and q with equal perpendicular distances d between them.
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In a coordinate system or on a grid, the horizontal axis (X-axis) and the vertical axis (Y-axis) are always perpendicular to each other, meeting at the origin (0,0)(0,0).

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A perpendicular bisector is a line that is perpendicular to a segment and passes through its midpoint, dividing the segment into two equal halves.

📐Formulae

If l⊥m, then the angle between them =90∘\text{If } l \perp m, \text{ then the angle between them } = 90^\circ

If l∥m, then the perpendicular distance d is constant.\text{If } l \parallel m, \text{ then the perpendicular distance } d \text{ is constant.}

∠ABC+∠CBD=180∘ (Linear pair on a straight line)\angle ABC + \angle CBD = 180^\circ \text{ (Linear pair on a straight line)}

💡Examples

Problem 1:

If a line XYXY is perpendicular to line PQPQ at point OO, what is the measure of ∠XOP\angle XOP?

Solution:

Given XY⊥PQXY \perp PQ, the angle formed at the intersection point OO must be a right angle. Therefore, ∠XOP=90∘\angle XOP = 90^\circ.

Explanation:

By definition, perpendicular lines meet at an angle of 90∘90^\circ.

Problem 2:

Two parallel lines ll and mm are intersected by a transversal. If the interior angles on the same side of the transversal are (2x+10)∘(2x + 10)^\circ and 70∘70^\circ, find the value of xx.

Solution:

Interior angles on the same side of a transversal are supplementary (sum up to 180∘180^\circ). (2x+10)+70=180(2x + 10) + 70 = 180 2x+80=1802x + 80 = 180 2x=180−802x = 180 - 80 2x=1002x = 100 x=1002=50x = \frac{100}{2} = 50 Calculation for 180−80180 - 80: 180−80100\begin{array}{r} 180 \\ -80 \\ \hline 100 \end{array}

Explanation:

We use the property that consecutive interior angles formed by a transversal intersecting parallel lines are supplementary.

Problem 3:

In a rectangle ABCDABCD, identify which sides are perpendicular to each other.

Solution:

In rectangle ABCDABCD, the adjacent sides meet at 90∘90^\circ. Therefore:

  1. AB⊥BCAB \perp BC
  2. BC⊥CDBC \perp CD
  3. CD⊥DACD \perp DA
  4. DA⊥ABDA \perp AB

Explanation:

A rectangle is a quadrilateral where all internal angles are 90∘90^\circ, meaning every pair of adjacent sides is perpendicular.

Problem 4:

In the given figure, if line ABAB is perpendicular to line CDCD at point MM, and the angle ∠AMC\angle AMC is represented as (3x−15)∘(3x - 15)^{\circ}, find the value of xx.

Lines AB and CD intersecting perpendicularly at point M.

Solution:

  1. Since AB⊥CDAB \perp CD, the angle between them is 90∘90^{\circ}.
  2. Therefore, ∠AMC=90∘\angle AMC = 90^{\circ}.
  3. Set up the equation: 3x−15=903x - 15 = 90.
  4. 3x=90+153x = 90 + 15.
  5. 3x=1053x = 105.
  6. x=1053=35x = \frac{105}{3} = 35.

Explanation:

Because the lines are perpendicular, any of the four angles formed at the intersection is 90∘90^{\circ}. Solving the linear equation gives the value of the unknown variable.

Problem 5:

Given a square PQRSPQRS. Identify the pairs of adjacent sides that are perpendicular and the pairs of opposite sides that are parallel.

A square PQRS showing perpendicular adjacent sides and parallel opposite sides.

Solution:

  1. In a square, all internal angles are 90∘90^{\circ}.
  2. Perpendicular pairs (adjacent): PQ⊥QRPQ \perp QR, QR⊥RSQR \perp RS, RS⊥SPRS \perp SP, and SP⊥PQSP \perp PQ.
  3. Parallel pairs (opposite): PQ∥SRPQ \parallel SR and PS∥QRPS \parallel QR.

Explanation:

By definition, a square has four right angles, making adjacent sides perpendicular, and two pairs of equal, parallel opposite sides.