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Lines and Angles - Lines Parallel to the Same Line

Grade 9CBSE

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

🔑Concepts

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Transitivity of Parallelism: Lines which are parallel to the same line are parallel to each other. For instance, if line ll is parallel to line mm, and line nn is also parallel to line mm, then ll and nn will never meet, making them parallel. Visually, imagine three horizontal slats of a window blind; if the top and bottom are parallel to the middle, they are parallel to each other.

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Theorem 6.6: This theorem formally states that lines which are parallel to the same line are parallel to each other. This is a fundamental property used to prove parallelism in complex geometric figures involving multiple sets of lines, often depicted as a series of non-intersecting layers.

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Verification via Transversal: When a transversal line tt intersects three lines l,m,l, m, and nn where lparallelml \\parallel m and nparallelmn \\parallel m, the corresponding angles formed at the intersection of tt with ll and nn will be equal. Visually, this creates matching 'F-shapes' at each junction along the transversal.

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Alternate Interior Angle Equality: When lines ll and nn are parallel to a third line mm, any transversal intersecting them creates equal alternate interior angles between ll and nn. This relationship is often visualized as identical 'Z-shapes' formed between the outermost parallel lines.

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Consecutive Interior Angles: If lparallelml \\parallel m and nparallelmn \\parallel m, then lparallelnl \\parallel n, and the interior angles on the same side of a transversal between ll and nn sum to 180circ180^{\\circ}. This forms a 'C-shape' or 'U-shape' where the two interior corner angles are supplementary.

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Non-intersection Principle: Because parallel lines maintain a constant perpendicular distance, if two distinct lines are both parallel to a third line, they cannot have a common point and therefore cannot intersect each other, preserving parallelism across the set.

📐Formulae

lparallelmtextandnparallelmimplieslparallelnl \\parallel m \\text{ and } n \\parallel m \\implies l \\parallel n

angle1=angle2text(CorrespondingAngles)\\angle 1 = \\angle 2 \\text{ (Corresponding Angles)}

angle3=angle4text(AlternateInteriorAngles)\\angle 3 = \\angle 4 \\text{ (Alternate Interior Angles)}

angle5+angle6=180circtext(Co−interiorAngles)\\angle 5 + \\angle 6 = 180^{\\circ} \\text{ (Co-interior Angles)}

💡Examples

Problem 1:

In the figure, ABparallelCDAB \\parallel CD and CDparallelEFCD \\parallel EF. Given that the ratio of angles y:z=3:7y:z = 3:7, where yy is an angle at CDCD and zz is an alternate interior angle at EFEF relative to the transversal, find the value of angle xx at ABAB.

Solution:

  1. Since ABparallelCDAB \\parallel CD and CDparallelEFCD \\parallel EF, then ABparallelEFAB \\parallel EF because lines parallel to the same line are parallel to each other.
  2. For ABparallelEFAB \\parallel EF, the alternate interior angles xx and zz are equal, so x=zx = z.
  3. Since ABparallelCDAB \\parallel CD, the sum of co-interior angles is supplementary: x+y=180circx + y = 180^{\\circ}.
  4. Substituting x=zx = z into the equation, we get z+y=180circz + y = 180^{\\circ}.
  5. Given the ratio y:z=3:7y:z = 3:7, let y=3ky = 3k and z=7kz = 7k.
  6. 3k+7k=180circimplies10k=180circimpliesk=18circ3k + 7k = 180^{\\circ} \\implies 10k = 180^{\\circ} \\implies k = 18^{\\circ}.
  7. Therefore, x=z=7times18circ=126circx = z = 7 \\times 18^{\\circ} = 126^{\\circ}.

Explanation:

The problem is solved by using the transitivity property to establish that the first and third lines are parallel. By then applying the properties of alternate interior angles and co-interior angles, we solve for the unknown using the given ratio.

Problem 2:

Prove that if line lparallelml \\parallel m and line nparallelln \\parallel l, then line mparallelnm \\parallel n.

Solution:

  1. Draw a transversal tt intersecting lines l,m,l, m, and nn.
  2. Let angle1,angle2,\\angle 1, \\angle 2, and angle3\\angle 3 be the corresponding angles formed by tt with l,m,l, m, and nn respectively.
  3. Since lparallelml \\parallel m, then angle1=angle2\\angle 1 = \\angle 2 (Corresponding Angles Axiom).
  4. Since nparallelln \\parallel l, then angle3=angle1\\angle 3 = \\angle 1 (Corresponding Angles Axiom).
  5. Using the transitive property of equality, if angle1=angle2\\angle 1 = \\angle 2 and angle3=angle1\\angle 3 = \\angle 1, then angle2=angle3\\angle 2 = \\angle 3.
  6. Because the corresponding angles angle2\\angle 2 and angle3\\angle 3 are equal, the lines mm and nn must be parallel to each other.

Explanation:

This geometric proof relies on a transversal to create corresponding angles, then uses the axiomatic property of parallel lines and numerical transitivity to prove that the final two lines are parallel.